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In this paper, we consider a mean field game model inspired by crowd motion where agents aim to reach a closed set, called target set, in minimal time. Congestion phenomena are modeled through a constraint on the velocity of an agent that depends on the average density of agents around their position. The model is considered in the presence of state constraints: roughly speaking, these constraints may model walls, columns, fences, hedges, or other kinds of obstacles at the boundary of the domain which agents cannot cross. After providing a more detailed description of the model, the paper recalls some previous results on the existence of equilibria for such games and presents the main difficulties that arise due to the presence of state constraints. Our main contribution is to show that equilibria of the game satisfy a system of coupled partial differential equations, known mean field game system, thanks to recent techniques to characterize optimal controls in the presence of state constraints. These techniques not only allow to deal with state constraints but also require very few regularity assumptions on the dynamics of the agents. [ABSTRACT FROM AUTHOR]

This paper is devoted to a sharp internal/boundary observability inequality for a hyperbolic system with a zero order potential. For this purpose, we first establish a new Carleman estimate for hyperbolic operator in H1-norm. Based on this Carleman estimate and a modified auxiliary optimal control problem, we obtain Carleman estimate for hyperbolic operator in L2-norm. Then, by virtue of a modified energy estimate and a delicate treatment of the observation region, we obtain an internal observability estimate with the observability constant of the order exp (C‖q‖2/3L∞(Q;ℝN×N)), with q the potential involved in the system. We also address the same problem for boundary observation. Compared with the related results in the literature, the main contributions of this paper are the observability constant is sharper, the waiting time T is shorter and the internal (or boundary) observation domain is smaller. [ABSTRACT FROM AUTHOR]

We study the energy decay of a damped Euler-Bernoulli beam which is subject to a point-wise feedback force representing a Shape Memory Alloy (SMA) cable. The problem we consider is that of the paper by [Liu et al. Int. J. Struct. Stab. Dyn.17 (2017) 1750076] but, for simplicity, our mod-elization does not take into account the additional stiffness term they considered. An explicit expression is given for the resolvent of the underlying operator as well as its eigenvalues and eigenfunctions. We show the exponential decay of the energy. The fastest decay rate is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup since we prove the existence of a Riesz basis. To the question "Do Shape Memory Alloy cables restrain the vibrations of girder bridges?", the experiments in [Liu et al. Int. J. Struct. Stab. Dyn.17 (2017) 1750076] answer positively. Our study does not allow to give a definite answer yet. The only presence of these cables may not be enough. Some physical parameters have to be chosen carefully. [ABSTRACT FROM AUTHOR]

In this paper we focus on a general optimal control problem involving a dynamical system described by a nonlinear Caputo fractional differential equation of order 0 < α ≤ 1, associated to a general Bolza cost written as the sum of a standard Mayer cost and a Lagrange cost given by a Riemann-Liouville fractional integral of order β ≥ α. In addition the present work handles general control and mixed initial/final state constraints. Adapting the standard Filippov's approach based on appropriate compactness assumptions and on the convexity of the set of augmented velocities, we give an existence result for at least one optimal solution. Then, the major contribution of this paper is the statement of a Pontryagin maximum principle which provides a first-order necessary optimality condition that can be applied to the fractional framework considered here. In particular, Hamiltonian maximization condition and transversality conditions on the adjoint vector are derived. Our proof is based on the sensitivity analysis of the Caputo fractional state equation with respect to needle-like control perturbations and on Ekeland's variational principle. The paper is concluded with two illustrating examples and with a list of several perspectives for forthcoming works. [ABSTRACT FROM AUTHOR]

This paper studies an equivalence theorem for three different kinds of optimal control problems, which are optimal time control problems, optimal norm control problems, and optimal target control problems. The controlled systems in this paper are internally controlled linear heat equations with memory. [ABSTRACT FROM AUTHOR]

In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing appears in the study of stability and safety, typically using Lyapunov and barrier functions, respectively. The results in this paper present infinitesimal conditions that do not require any knowledge about the solutions to the system. Results under different regularity properties of the considered scalar function are provided. This includes when the scalar function is lower semicontinuous, locally Lipschitz and regular, or continuously differentiable. [ABSTRACT FROM AUTHOR]

In this paper, we study an optimal control problem for a nonlinear system of reaction–diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in H. Garcke, et al. [Math. Model. Methods Appl. Sci.26 (2016) 1095–1148]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [P. Colli, et al. To appear in: Appl. Math. Optim. (2019)] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L1−norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [P. Colli, et al. To appear in: Appl. Math. Optim. (2019)], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class of multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using an adjoint calculus for the inverse of the non-linear and non-differentiable directional derivative of the solution map of the considered PDE, we extend the concept of strong stationarity to the multiobjective setting and demonstrate that the properties of weak and proper Pareto stationarity can also be characterized by suitable multiplier systems that involve both primal and dual quantities. The established optimality conditions imply in particular that Pareto stationary points possess additional regularity properties and that mollification approaches are – in a certain sense – exact for the studied problem class. We further show that the obtained results are closely related to rather peculiar hidden regularization effects that only reveal themselves when the control is eliminated and the problem is reduced to the state. This observation is also new for the case of a single objective function. The paper concludes with numerical experiments that illustrate that the derived optimality systems are amenable to numerical solution procedures. [ABSTRACT FROM AUTHOR]

This paper develops an analytic framework to design both stress-controlled and displacement-controlled T-periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t↦(e(t), p(t)), where ei(t) is the elastic elongation and pi(t) is the relaxed length of spring i, defined on [t0, ∞) by the initial condition (e(t0), p(t0)). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C(t) in a vector space E of dimension d, it becomes natural to expect (based on a result by Krejci) that the elastic component t↦e(t) always converges to a T-periodic function as t →∞. The achievement of this paper is in spotting a class of loadings where the Krejci's limit doesn't depend on the initial condition (e(t0), p(t0)) and so all the trajectories approach the same T-periodic regime. The proposed class of sweeping processes is the one for which the normals of any d different facets of the moving polyhedron C(t) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normals of any d different facets of the moving polyhedron C(t) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem. [ABSTRACT FROM AUTHOR]

A regularity result for free-discontinuity energies defined on the space SBVp(·) of special functions of bounded variation with variable exponent is proved, under the assumption of a log-Hölder continuity for the variable exponent p(x). Our analysis expands on the regularity theory for minimizers of a class of free-discontinuity problems in the nonstandard growth case. This may be seen as a follow-up of the paper N. Fusco et al., J. Convex Anal. 8 (2001) 349-367, dealing with a constant exponent. [ABSTRACT FROM AUTHOR]

In this paper, we study an optimal control problem of partially observed mean-field type stochastic control system with Markov chain in progressive structure. The control variable is allowed to enter the diffusion term of the state process and the drift term of the observation process. The control domain need not be convex. In our model, the cost functional and the observation are also of mean-field type. By virtue of a special spike variation, the related stochastic maximum principle has been obtained. The stochastic maximum principle in progressive structure is essentially different from the classical case. [ABSTRACT FROM AUTHOR]

In this paper, by isometrically embedding (Sn+1, gSn+1) into Rn+2, and using nonlinear analysis on the codimension-2 graphs, we will show that the level-sets of the distance function from the boundary of any open set in sphere, are C1,1-rectifiable. As a by-product, we establish a Heintze-Karcher-type inequality on sphere. [ABSTRACT FROM AUTHOR]

In this paper, we study a Dirichlet problem for noncoercive nonlinear elliptic equations with first order term in an unbounded domain. We obtain Stampacchia type existence, regularity and uniqueness results, when the singular drift term is controlled through a function in a suitable functional space, strictly containing Lebesgue one. The main tools are a weak maximum principle together with some a priori estimates proved by contradiction. [ABSTRACT FROM AUTHOR]

The saturated boundary stabilization problem for quasi-linear hyperbolic systems of balance laws is considered under H2-norm in this paper, where the boundary conditions of the system are subject to actuator saturations. The resulting closed-loop system is proven to be locally exponentially stable with respect to the steady states in the presence of saturations. To this end, the sector nonlinearity model is introduced to deal with the saturation term, and then sufficient conditions for ensuring the locally exponential stability are established in terms of a set of matrix inequalities by employing the Lyapunov function method along with a sector condition. Furthermore, these results are applied to the stabilization of the two-lane traffic flow dynamic represented by Lighthill–Whitham–Richards (LWR) model. By utilizing variable speed limit (VSL) devices, a saturated boundary feedback controller is designed to stabilize the two-lane traffic flow, and the exponential convergence of the quasi-linear traffic flow system in H2 sense is validated by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper addresses the qualitative theory of mixed-order positive linear coupled systems with bounded or unbounded delays. First, we introduce a general result on the existence and uniqueness of solutions to mixed-order linear systems with time-varying delays. Next, we obtain necessary and sufficient criteria which characterize the positivity of mixed-order delay linear coupled systems. Our main contributions are in Section 5. More precisely, by using a smoothness property of solutions to fractional differential equations and developing a new appropriated comparison principle for solutions to mixed-order delay positive systems, we prove the attractivity of mixed-order non-homogeneous linear positive coupled systems under the impact of bounded or unbounded delays. We also establish a necessary and sufficient condition to| ensure the stability of homogeneous systems. As a consequence of these results, we show the smallest asymptotic bound of solutions to mixed-order delay positive non-homogeneous linear coupled systems where disturbances are continuous and bounded. Finally, we provide numerical simulations to illustrate the proposed theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study an inverse problem of estimating three discontinuous parameters in a double phase implicit obstacle problem with multivalued terms and mixed boundary conditions which is formulated by a regularized optimal control problem. Under very general assumptions, we introduce a multivalued function called a parameter-to-solution map which admits weakly compact values. Then, by employing the Aubin-Cellina convergence theorem and the theory of nonsmooth analysis, we prove that the parameter-to-solution map is bounded and continuous in the sense of Kuratowski. Finally, a generalized regularization framework for the inverse problem is developed and a new existence theorem is provided. [ABSTRACT FROM AUTHOR]

Higher order necessary conditions for a minimizer of an optimal control problem are generally obtained for systems whose dynamics is continuously differentiable in the state variable. Here, by making use of the notion of set-valued Lie bracket, we obtain a Goh-type condition for a control affine system with Lipschitz continuous dynamics and unbounded controls. In order to manage the simultaneous lack of smoothness of the adjoint equation and of the Lie bracket-like variations we make use of the notion of Quasi Differential Quotient. We conclude the paper with a worked out example where the established higher order condition is capable to rule out the optimality of a control verifying the standard maximum principle. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the problem of determining perfectly insulating regions (cavities) from one boundary measurement in a nonlinear elliptic equation arising from cardiac electrophysiology. Based on the results obtained in [9] we propose a new reconstruction algorithm based on Γ-convergence. The relevance and applicability of this approach are then shown through several numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is concerned with a backward stochastic linear-quadratic (LQ, for short) optimal control problem with deterministic coefficients. The weighting matrices are allowed to be indefinite, and cross-product terms in the control and state processes are presented in the cost functional. Based on a Hilbert space method, necessary and sufficient conditions are derived for the solvability of the problem, and a general approach for constructing optimal controls is developed. The crucial step in this construction is to establish the solvability of a Riccati-type equation, which is accomplished under a fairly weak condition by investigating the connection with forward stochastic LQ optimal control problems. [ABSTRACT FROM AUTHOR]

In this paper we provide a rate of convergence for periodic homogenization of Hamilton–Jacobi–Bellman equations with nonlocal diffusion. The result is based on the regularity of the associated effective problem, where convexity plays a crucial role. The necessary regularity estimates are made possible by a representation formula we obtain for the effective Hamiltonian, a result that has an independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we study a natural optimal control problem associated to the Paneitz obstacle problem on closed 4-dimensional Riemannian manifolds. We show the existence of an optimal control which is an optimal state and induces also a conformal metric with prescribed Q-curvature. We show also C∞-regularity of optimal controls and some compactness results for the optimal controls. In the case of the 4-dimensional standard sphere, we characterize all optimal controls. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotics of singularly perturbed phase-transition functionals of the form ℱk(u) = 1/εk∫Afk(푥,u,εk∇u)d푥, where u ∈ [0, 1] is a phase-field variable, εk > 0 a singular-perturbation parameter i.e., εk → 0, as k → +∞, and the integrands fk are such that, for every x and every k, fk(x, ·, 0) is a double well potential with zeros at 0 and 1. We prove that the functionals Fk Γ-converge (up to subsequences) to a surface functional of the form ℱ∞(u) = ∫Su∩Af∞(푥,휈u)dHn-1, where u ∈ BV(A; {0, 1}) and f∞ is characterised by the double limit of suitably scaled minimisation problems. Afterwards we extend our analysis to the setting of stochastic homogenisation and prove a Γ-convergence result for stationary random integrands. [ABSTRACT FROM AUTHOR]

In this paper, we study the small-time global null controllability of the generalized Burgers' equations yt + γ|y|γ-1yx — yxx = u(t) on the segment [0, 1]. The scalar control u(t) is uniform in space and plays a role similar to the pressure in higher dimension. We set a right Dirichlet boundary condition y(t, 1) = 0, and allow a left boundary control y(t, 0) = v(t). Under the assumption γ > 3/2 we prove that the system is small-time globally null controllable. Our proof relies on the return method and a careful analysis of the shape and dissipation of a boundary layer. [ABSTRACT FROM AUTHOR]

In this paper, we consider networks of N synaptically coupled excitatory-inhibitory neural modules, with N arbitrary. It has been argued that the connection strengths may slowly vary with respect to time and that they can actually be considered as inputs of the network. The problem that we are studying is which connection strengths should be modified (in other words, which connection strengths should be considered as inputs) in order to achieve flatness for the resulting control system. Flatness of the control network depends on the number of inputs and we show that if enough connection strengths (at least N) can be considered as inputs, then the control network is flat without structural conditions. If the number of inputs is smaller than N. then flatness imposes particular configurations of the interactions between the subnetworks. Our main contribution is to identify, analyze and characterize several flat configurations in the latter case. [ABSTRACT FROM AUTHOR]

This paper is addressed to establishing controllability and observability for some forward linear stochastic complex degenerate/singular Ginzburg-Landau equations. It is sufficient to establish appropriate observability inequalities for the corresponding backward and forward equations. The key is to prove the Carleman estimates of the forward and backward linear stochastic complex degenerate/singular Ginzburg-Landau operators. Compared with the existing deterministic results, it is necessary to overcome the difficulties caused by some complex coefficients and random terms. The results obtained cover those of deterministic cases and generalize those of stochastic degenerate parabolic equations. Moreover, the limit behavior of the coefficients in the equation is discussed. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of the optimal control of an ensemble of affine-control systems. After proving the well-posedness of the minimization problem under examination, we establish a Γ-convergence result that allows us to substitute the original (and usually infinite) ensemble with a sequence of finite increasing-in-size sub-ensembles. The solutions of the optimal control problems involving these sub-ensembles provide approximations in the L2-strong topology of the minimizers of the original problem. Using again a Γ-convergence argument, we manage to derive a Maximum Principle for ensemble optimal control problems with end-point cost. Moreover, in the case of finite sub-ensembles, we can address the minimization of the related cost through numerical schemes. In particular, we propose an algorithm that consists of a subspace projection of the gradient field induced on the space of admissible controls by the approximating cost functional. In addition, we consider an iterative method based on the Pontryagin Maximum Principle. Finally, we test the algorithms on an ensemble of linear systems in ℝ2. [ABSTRACT FROM AUTHOR]

In this paper we deal with the controllability properties of a system of m coupled Stokes systems or m coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control a m × N state (corresponding to the velocities of the fluids) by N — 1 distributed scalar controls. The proof of the controllability of the coupled Stokes systems is based on a Carleman estimate for the adjoint system. The local null-controllability of the coupled Navier-Stokes systems is then obtained by means of the source term method and a Banach fixed point. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot–Carathéodory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of Γ-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle. [ABSTRACT FROM AUTHOR]

In this paper, we study the quantitative analyticity and observability inequality for solutions of fractional order parabolic equations with space-time dependent potentials in ℝn. We first obtain a uniformly lower bound of analyticity radius of the spatial variable for the above solutions with respect to the time variable. Next, we prove a globally Hölder-type interpolation inequality on a thick set, which is based on a propagation estimate of smallness for analytic functions. Finally, we establish an observability inequality from a thick set in ℝn, by utilizing a telescoping series method. [ABSTRACT FROM AUTHOR]

This paper is concerned with a discounted optimal control problem of partially observed forward-backward stochastic systems with jumps on infinite horizon. The control domain is convex and a kind of infinite horizon observation equation is introduced. The uniquely solvability of infinite horizon forward (backward) stochastic differential equation with jumps is obtained and more extended analyses, especially for the backward case, are made. Some new estimates are first given and proved for the critical variational inequality. Then a maximum principle is obtained by introducing some infinite horizon adjoint equations whose uniquely solvabilities are guaranteed necessarily. Finally, some comparisons are made with two kinds of representative infinite horizon stochastic systems and their related optimal controls. [ABSTRACT FROM AUTHOR]

In this paper, we consider a transmission wave equation in N embedded domains with multiple interfaces of discontinuous coefficients in ℝ2. We study the global stability in determining the source term from a one-measurement data of wavefield velocity in a subboundary over a time interval. We prove the stability estimate for this inverse source problem by a combination of the local hyperbolic/elliptic Carleman estimates and the Fourier-Bros-Iagolniter transformation. Our method could be generalized to general dimensions since the weight functions and Carleman estimates are independent of the dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of nonlinear (in particular, saturated) stabilization of the high-dimensional wave equation with Dirichlet boundary conditions. The wave dynamics are subject to a dissipative nonlinear velocity feedback and generate a strongly continuous semigroup of contractions on the optimal energy space L2(Ω) × H−1(Ω). It is first proved that any solution to the closed-loop equations converges to zero in the aforementioned topology. Secondly, under the condition that the feedback nonlinearity has linear growth around zero, polynomial energy decay rates are established for solutions with smooth initial data. This constitutes new Dirichlet counterparts to well-known results pertaining to nonlinear stabilization in H1(Ω) × L2(Ω) of the wave equation with Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints in Banach spaces. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications to shape optimization. This abstract framework yields practical formulae to compute the derivative of a shape functional, the material derivative of the state, and the adjoint state. Furthermore, it allows to gain insight on the duality between the material derivative of the state and the adjoint state. We show several applications of this method to the computation of distributed shape derivatives for problems involving linear elliptic, nonlinear elliptic, parabolic PDEs and distributions. We also compare our approach with other techniques for computing shape derivatives including the material derivative method and the averaged adjoint method. [ABSTRACT FROM AUTHOR]

This paper concerns the null control of quasi-linear parabolic systems where the diffusion coefficient depends on the gradient of the state variable. In our main theoretical result, with some assumptions on the regularity and growth of the diffusion coefficient and regular initial data, we prove that local null controllability holds. To this purpose, we consider the null controllability problem for the linearized system, we deduce new estimates on the control and the state and, then, we apply a Local Inversion Theorem. We also formulate an iterative algorithm of the quasi-Newton kind for the computation of a null control and an associated state. We apply this method to some numerical approximations of the problem and illustrate the results with several experiments. [ABSTRACT FROM AUTHOR]

In this paper we consider optimal sampled-data control problems on time scales with inequality state constraints. A Pontryagin maximum principle is established, extending to the state constrained case existing results in the time scale literature. The proof is based on the Ekeland variational principle and on the concept of implicit spike variations adapted to the time scale setting. The main result is then applied to continuous-time min-max optimal sampled-data control problems, and a maximal velocity minimization problem for the harmonic oscillator with sampled-data control is numerically solved for illustration. [ABSTRACT FROM AUTHOR]

In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied in this paper: the weakly degenerate case, that is, if the reciprocal of the diffusion coefficient is summable, and the strongly degenerate case, that is, if that reciprocal isn't summable. In our main result we show that the above systems can be steered from an initial continuous state that admits a finite number of points of sign change to a target state with the same number of changes of sign in the same order. Our method uses a recent technique introduced for uniformly parabolic equations employing the shifting of the points of sign change by making use of a finite sequence of initial-value pure diffusion problems. Our interest in degenerate reaction-diffusion equations is motivated by the study of some energy balance models in climatology (see, e.g., the Budyko-Sellers model) and some models in population genetics (see, e.g., the Fleming-Viot model). [ABSTRACT FROM AUTHOR]

In this paper, we study the observability and controllability of wave equations coupled by first or zero order terms on a compact manifold. We adopt the approach in Dehman-Lebeau's paper [B. Dehman and G. Lebeau, SIAM J. Control Optim.48 (2009) 521–550.] to prove that: the weak observability inequality holds for wave equations coupled by first order terms on compact manifold without boundary if and only if a class of ordinary differential equations related to the symbol of the first order terms along the Hamiltonian flow are exactly controllable. We also compute the higher order part of the observability constant and the observation time. By duality, we obtain the controllability of the dual control system in a finite co-dimensional space. This gives the full controllability under the assumption of unique continuation of eigenfunctions. Moreover, these results can be applied to the systems of wave equations coupled by zero order terms of cascade structure after an appropriate change of unknowns and spaces. Finally, we provide some concrete examples as applications where the unique continuation property indeed holds. [ABSTRACT FROM AUTHOR]

Piezoelectric materials can be controlled with current (or charge) as the electrical input, instead of voltage. The main purpose of this paper is to derive the governing equations for a current-controlled piezo-electric beam and to investigate stabilizability. The magnetic permeability in piezo-electric materials is generally neglected in models. However, it has a significant qualitative effect on properties of the control system such as stabilizability. Besides the consideration of current control, there are several new aspects to the model. Most importantly, a fully dynamic magnetic model is included. Also, electrical potential and magnetic vector potential are chosen to be quadratic-through thickness to include the induced effects of the electromagnetic field. Hamilton's principle is used to derive a boundary value problem that models a single piezo-electric beam actuated by a current (or charge) source at the electrodes. Two sets of decoupled system of partial differential equations are obtained; one for stretching of the beam and another one for bending motion. Since current (or charge) controller only affects the stretching motion, attention is focused on control of the stretching equations in this paper. It is shown that the Lagrangian of the beam is invariant under certain transformations. A Coulomb type gauge condition is used. This gauge condition decouples the electrical potential equation from the equations of the magnetic potential. A semigroup approach is used to prove that the Cauchy problem is well-posed. Unlike voltage actuation, a bounded control operator in the natural energy space is obtained. The paper concludes with analysis of stabilizability and comparison with other actuation approaches and models. [ABSTRACT FROM AUTHOR]

In this paper, we address the question of the construction of a nearly time optimal feedback law for a minimum time optimal control problem, which is robust with respect to internal and external perturbations. For this purpose we take as starting point an optimal synthesis, which is a suitable collection of optimal trajectories. The construction we exhibit depends exclusively on the initial data obtained from the optimal feedback which is assumed to be known. [ABSTRACT FROM AUTHOR]

In this paper we derive the continuum limit of a multiple-species, interacting particle system by proving a Γ-convergence result on the interaction energy as the number of particles tends to infinity. As the leading application, we consider n edge dislocations in multiple slip systems. Since the interaction potential of dislocations has a logarithmic singularity at zero with a sign that depends on the orientation of the slip systems, the interaction energy is unbounded from below. To make the minimization problem of this energy meaningful, we follow the common approach to regularise the interaction potential over a length-scale δ > 0. The novelty of our result is that we leave the type of regularisation general, and that we consider the joint limit n →∞ and δ → 0. Our result shows that the limit behaviour of the interaction energy is not affected by the type of the regularisation used, but that it may depend on how fast the size (i.e., δ) decays as n →∞. [ABSTRACT FROM AUTHOR]

In this paper we prove the time-domain boundedness for noise-to-state exponentially stable systems, and further make an estimation of its lower bound function, which allows to answer the question that how long the solution of a stochastic noise-to-state exponentially stable system stays in the domain of attraction and what happens with it if it escapes from this region for a while. The results will complement the probability-domain boundedness of noise-to-state exponentially stable systems, and provide a new insight into noise-to-state exponential stability. [ABSTRACT FROM AUTHOR]

This paper is devoted to the stabilization for a class of uncertain linearized Ginzburg–Landau equations (GLEs). The distinguishing feature of such system is the presence of serious uncertainties which enlarge the scope of the systems whereas challenge the control problem. Therefore, certain dynamic compensation mechanisms are required to overcome the uncertainties of system. Motivated by the related literature, the original complex-valued GLEs are transformed into a class of real-valued coupled parabolic systems with serious uncertainties and distinctive characteristics. For this, two classes of identifiers respectively based on passive and swapping identifiers are first introduced to design parameter dynamic compensators. Then, by combining infinite-dimensional backstepping method with the dynamic compensators, two adaptive state-feedback controllers are constructed which guarantee all the closed-loop system states are bounded while the original system states converge to zero. A numerical example is provided to validate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Adaptive boundary stabilization is investigated for a class of PDE-ODE cascade systems with general uncertain harmonic disturbances. The essential difference between this paper and the existing related literature is the presence of the uncertain disturbances belonging to an unknown interval, which makes the problem unsolved so far. Motivated by the existing related literature, the paper develops the adaptive boundary stabilization for the PDE-ODE cascade system in question. First, an adaptive boundary feedback controller is constructed in two steps by adaptive and Lyapunov techniques. Then, it is shown that the resulting closed-loop system is well-posed and asymptotically stable, by the semigroup approach and LaSalle's invariance principle, respectively. Moreover, the parameter estimates involved in the designed controller are shown to ultimately converge to their own real values. Finally, the effectiveness of the proposed method is illustrated by a simulation example. [ABSTRACT FROM AUTHOR]

In the well-known paper [A. Bahri and J.M. Coron, Commun. Pure Appl. Math. 41 (1988) 253-294], Bahri and Coron develop the theory of critical points at innity and find the solutions of Yamabe problem via Morse theory. This is a very delicate problem because of the lack of compactness caused by the invariance under the conformal group. To obtain the desired results, one needs a careful analysis on the change of the topology of the level sets. In this work, the author continues to use these ideas and give a preliminary study of the topological features for the Yamabe sign-changing variational problem on domains of ℝ³ or on spheres 핊³. One of key points consists to understand the Morse relations at innity based on the expansion of the energy functional in a neighborhood of innity. In particular, one study weather the relation ∂∞?∂∞ = 0 holds where ∂∞ is the intersection operator at innity. Although I could not understand completely the details, I believe such study is very delicate and the ideas and techniques developed could be also useful in the others context, in particular, some conformal invariant problems like Yang-Mills equations and harmonic maps. I recommend strongly the publication of the paper. [ABSTRACT FROM AUTHOR]

This paper studies a vertical powered descent problem in the context of planetary landing, considering glide-slope and thrust pointing constraints and minimizing any final cost. In a first time, it proves the Max-Min-Max or Max-Singular-Max form of the optimal control using the Pontryagin Maximum Principle, and it extends this result to a problem formulation considering the effect of an atmosphere. It also shows that the singular structure does not appear in generic cases. In a second time, it theoretically analyzes the optimal trajectory for a more specific problem formulation to show that there can be at most one contact or boundary interval with the state constraint on each Max or Min arc. [ABSTRACT FROM AUTHOR]

This paper investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping associated with the system of first order necessary optimality conditions. Preliminary results concerning the differentiability of the functions involved are established, especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed, and under weaker assumptions than the ones previously considered in the literature on control constrained elliptic problems. Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization. [ABSTRACT FROM AUTHOR]

In this paper, we study a reverse isoperimetric inequality for planar convex bodies whose radius of curvature is between two positive numbers 0 ≤ α ≤ β, called (α, β)-convex bodies. We show that among planar (α, β)-convex bodies of fixed perimeter, the extremal shape is a domain whose boundary is composed by two arcs of circles of radius α joined by two arcs of circles of radius β. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two families of fully coupled linear Forward-Backward Stochastic Differential Equations (FBSDEs) and its applications to optimal Linear Quadratic (LQ) problems. Within these families, one could get same well-posedness of FBSDEs with totally different coefficients. A family of FBSDEs is proved to be equivalent with respect to the Unified Approach. Thus one could get well-posedness of whole family once a member exists a unique solution. Another equivalent family of FBSDEs are investigated by introducing a linear transformation method. Owing to the coupling structure between forward and backward equations, it leads to a highly interdependence in solutions. We are able to decouple FBSDEs into partial coupling, by virtue of linear transformation, without losing the existence and uniqueness to solutions. Moreover, owing to non-degeneracy of transformation matrix, the solution to original FBSDEs is totally determined by solutions of FBSDEs after transformation. In addition, an example of optimal LQ problem is presented to illustrate. [ABSTRACT FROM AUTHOR]

This paper is concerned with controllability and stabilization properties of the dispersion generalized Benjamin equation on the periodic domain T. First, by assuming the control input acts on all the domain, the system is proved to be globally exactly controllable in the Sobolev space Hsp(핋), with s ≥ 0. Second, by providing a locally-damped term added to the equation as a feedback law, it is shown that the resulting equation is globally well-posed and locally exponentially stabilizable in the space L2p(핋). The main ingredient to prove the global well-posedness is the introduction of the dissipation-normalized Bourgain spaces which allows one to gain smoothing properties simultaneously from the dissipation and dispersion present in the equation. Finally, the local exponential stabilizability result is accomplished taking into account the decay of the associated semigroup combined with the fixed point argument. [ABSTRACT FROM AUTHOR]

In recent years there have been many in-depth researches on the boundary controllability and boundary synchronization for coupled systems of wave equations with various types of boundary conditions. In order to extend the study of synchronization from wave equations to a much larger range of hyperbolic systems, in this paper we will define and establish the exact boundary synchronization for the first order linear hyperbolic system based on previous work on its exact boundary controllability. The determination and estimate of exactly synchronizable states and some related problems are also discussed. This work can be applied to a great deal of diverse systems, and a new perspective to study the synchronization problem for the coupled system of wave equations can be also provided. [ABSTRACT FROM AUTHOR]