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In this paper, we obtain some second order necessary conditions for optimal control problems of stochastic evolution equations in infinite dimensions. The control acts on both the drift and diffusi...

International audience; The purpose of this paper is to establish first and second order necessary optimality conditions for optimal control problems of stochastic evolution equations with control and state constraints. The control acts both in the drift and diffusion terms and the control region is a nonempty closed subset of a separable Hilbert space. We employ some classical set-valued analysis tools and theories of the transposition solution of vector-valued backward stochastic evolution equations and the relaxed-transposition solution of operator-valued backward stochas-tic evolution equations to derive these optimality conditions. The correction part of the second order adjoint equation, which does not appear in the first order optimality condition, plays a fundamental role in the second order optimality condition.

We study linear-quadratic optimal control problems with time-inconsistent cost functionals for stochastic evolution equations in a Hilbert space. A closed-loop equilibrium strategy is introduced fo...

This paper is devoted to studying controllability problems of linear stochastic control systems with controls only in the drift terms. It is well known that such a system is not exactly controllabl...

In this chapter, we present an introduction to backward stochastic evolution equations (valued in Hilbert spaces), which appear naturally in the study of control problems for stochastic distributed parameter systems. In the case of natural filtration, by means of the Martingale Representation Theorem, these equations are proved to be well-posed in the sense of mild solutions; while for the general filtration, using our stochastic transposition method, we also establish their well-posedness.

International audience; We establish some second order necessary conditions for optimal control problems of evolution equations involving final point equality and inequality constraints. Compared with the existing works, the main difference is due to the presence of end-point equality constraints. With such constraints, we cannot simply use the variational techniques since perturbations of a given control may be no longer admissible. We also cannot use the Ekeland’s variational principle, which is a first order variational principle, to obtain second order necessary conditions. Instead, we combine some inverse mapping theorems on metric spaces and second order linearization of data to obtain our results.

This chapter is devoted to presenting some results on controllability for forward and backward stochastic linear evolution equations with possibly unbounded control operators. By the duality argument, these controllability problems can be reduced to suitable observability for the dual equations. Explicit forms of controls for the controllability problems are also provided. Finally, the controllability of some forward stochastic evolution equations are shown to be equivalent to that of suitably chosen backward stochastic evolution equations.

In this chapter, we shall present several typical controlled stochastic evolution equations, including a new refined controlled stochastic wave equation in particular. Then, we shall give a quite general formulation of some typical control problems, such as controllability/observability problems and optimal control problems for stochastic distributed parameter systems.

This chapter is devoted to studying the null/approximate controllability and observability of stochastic parabolic systems.

In this chapter, we are concerned with linear quadratic optimal control problems (LQ problems for short) for stochastic evolution equations, in which the diffusion terms depend on the control variables and the coefficients are stochastic. In such a general setting, one has to introduce suitable operator-valued backward stochastic evolution equations (to characterize the optimal controls in the form of Pontryagin-type maximum principle or in the feedback forms), served as the second order adjoint equations or the Riccati type equations. As in the previous chapter, it is very difficult to show the existence of solutions to these equations. We shall use the stochastic transposition method to overcome this difficulty.

The aim of this notes is to give a concise introduction to control theory for systems governed by stochastic partial differential equations. We shall mainly focus on controllability and optimal control problems for these systems. For the first one, we present results for the exact controllability of stochastic transport equations, null and approximate controllability of stochastic parabolic equations and lack of exact controllability of stochastic hyperbolic equations. For the second one, we first introduce the stochastic linear quadratic optimal control problems and then the Pontryagin type maximum principle for general optimal control problems. It deserves mentioning that, in order to solve some difficult problems in this field, one has to develop new tools, say, the stochastic transposition method introduced in our previous works.

In this chapter, we shall prove that the usual stochastic wave equation, i.e., the classic wave equation perturbed by a term of Ito’s integral, is not exactly controllable even if the controls are effective everywhere in both drift and diffusion terms, which means that some key feature is ignored in this model. Then, by means of a global Carleman estimate, we establish the exact controllability of a refined stochastic wave equation with three controls. Moreover, we give a result about the lack of exact controllability, which shows that the action of three controls is necessary. Our analysis indicates that, at least from the view point of Control Theory, the new stochastic wave equation introduced in our work is more reasonable than the one in the existing literature.

This book is mainly addressed to studying the control problems governed by stochastic evolution equations. In this chapter, we shall present a short introduction to the well-posedness and regularity of solutions to this sort of equations, valued in Hilbert spaces.

The main purpose of this chapter is to derive first order necessary conditions, i.e., Pontryagin-type maximum principle for optimal controls of general nonlinear stochastic evolution equations in infinite dimensions, in which both the drift and the diffusion terms may contain the control variables, and the control regions are allowed to be nonconvex. In order to do this, quite different from the deterministic infinite dimensional setting and the stochastic finite dimensional case, people have to introduce a suitable operator-valued backward stochastic evolution equation, served as the second order adjoint equation. It is very difficult to prove the existence of solutions to this equation for the general case. Indeed, in the infinite dimensional setting, there exists no such a satisfactory stochastic integration/evolution equation theory (in the previous literatures) that can be employed to establish the well-posedness of such an equation.

We establish the well-posedness of operator-valued backward stochastic Lyapunov equations in infinite dimensions, in the sense of \begin{document}$ V $\end{document} -transposition solution and of relaxed transposition solution. As an application, we obtain a Pontryagin-type maximum principle for the optimal control of controlled stochastic evolution equations.

We analyze the averaged controllability properties of random evolution Partial Differential Equations. We mainly consider heat and Schr\"odinger equations with random parameters, although the problem is also formulated in an abstract frame. We show that the averages of parabolic equations lead to parabolic-like dynamics that enjoy the null-controllability properties of solutions of heat equations in an arbitrarily short time and from arbitrary measurable sets of positive measure. In the case of Schr\"odinger equations we show that, depending on the probability density governing the random parameter, the average may behave either as a conservative or a parabolic-like evolution, leading to controllability properties, in average, of very different kind., Comment: 51 pages, add some remarks according to the referee's comments

We study a linear quadratic optimal control problem for mean-field stochastic evolution equation with the assumption that all the coefficients concerned in the problem are deterministic. We show that the existence of optimal feedback operators is equivalent to that of regular solution to the system which is coupled by two Riccati equations and an explicit formula of the optimal feedback control operator is given via the regular solution. We also show that the mentioned Riccati equations admit a unique strongly regular solution when the cost functional is uniformly convex.

In this paper, we solve a local state observation problem for stochastic hyperbolic equations without boundary conditions, which is reduced to a local unique continuation property for these equations. This result is proved by a global Carleman estimate. As far as we know, this is the first result in this topic.

We study the closed-loop solvability of a stochastic linear quadratic optimal control problem for systems governed by stochastic evolution equations. This solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. At last, conditions ensuring the uniform convexity of the cost functional are discussed.

In this paper, motivated by the study of optimal control problems for infinite dimensional systems with endpoint state constraints, we introduce the notion of finite codimensional (exact/approximate) controllability. Some equivalent criteria on the finite codimensional controllability are presented. In particular, the finite codimensional exact controllability is reduced to deriving a G{\aa}rding type inequality for the adjoint system, which is new for many evolution equations. This inequality can be verified for some concrete problems (and hence applied to the corresponding optimal control problems), say the wave equations with both time and space dependent potentials. Moreover, under some mild assumptions, we show that the finite codimensional exact controllability of this sort of wave equations is equivalent to the classical geometric control condition., Comment: 34 pages

One of the fundamental issues in Control Theory is to design feedback controls. It is well-known that, the purpose of introducing Riccati equations in the study of deterministic linear quadratic control problems is exactly to construct the desired feedbacks. To date, the same problem in the stochastic setting is only partially well-understood. In this paper, we establish the equivalence between the existence of optimal feedback controls for the stochastic linear quadratic control problems with random coefficients and the solvability of the corresponding backward stochastic Riccati equations in a suitable sense. We also give a counterexample showing the nonexistence of feedback controls to a solvable stochastic linear quadratic control problem. This is a new phenomenon in the stochastic setting, significantly different from its deterministic counterpart.

This paper is addressed to an inverse stochastic hyperbolic problem with three unknowns, i.e., a random force intensity, an initial displacement, and an initial velocity. The global uniqueness for this inverse problem is proved by means of a new global Carleman estimate for the stochastic hyperbolic equation. It is found that both the formulation of stochastic inverse problems and the tools to solve them differ considerably from their deterministic counterpart. © 2015 Wiley Periodicals, Inc.

This paper is addressed to studying the exact controllability of stochastic Schrodinger equations by two controls. One is a boundary control and the other is an internal control in the diffusion term. By means of the duality argument, the control problem is converted into an observability problem for backward stochastic Schrodinger equations, and the desired observability estimate is obtained by a global Carleman estimate. At last, we give a result about the lack of exact controllability, which shows that the action of two controls is necessary.

We study the memory-type null controllability property for wave equations involving memory terms. The goal is not only to drive the displacement and the velocity (of the considered wave) to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. This memory-type null controllability problem can be reduced to the classical null controllability property for a coupled PDE-ODE system. The latter can be viewed as a degenerate system of wave equations, in which the velocity of propagation for the ODE component vanishes. This fact requires the support of the control to move to ensure the memory-type null controllability to hold, under the so-called Moving Geometric Control Condition. The control result is proved by duality by means of an observability inequality which employs measurements done on a moving observation open subset of the domain where the waves propagate., ´Nousétudions le contrôle vers l'´ equilibre de l'´ equations des ondes avec mémoire. Cecí exige non seulement de contrôler le déplacement et la vitesse de l'onde considérée, mais aussi d'assurer que le terme de mémoire atteigne l'´ etat d'´ equilibrè a un instant donné. Ceprobì eme de contrôlè a zéro de type mémoire peutêtrepeutˆpeutêtre réduitréduit`réduità unprobì eme de contrôlè a zéro pour un système couplé PDE-ODE. Ce dernier peutêtrepeutˆpeutêtre considéré comme un système dégénéré d'´ equations d'ondes, pour lequel la vitesse de propagation de la composante ODE est nulle. Pour aboutir au résultat, le support du contrôle doit se déplacer le long du domaine o` u les ondes se propagent, assurant ainsi, non seulement l'absorption des rayons classiques de l'´ equation des ondes, mais aussi labsorption des rayons qui ne se propagent pas et qui sont associésassociés`associésà l'ODE. On appelle cette condition : Condition de contrôle géométrique en mouvement. Le résultat de contrôle est prouvé moyennant une inégalité d'observabilité.

The classical Pontryagin maximum principle is a first-order necessary condition (FONC for short) for optimal controls of ordinary differential equations (ODEs for short). When the FONC degenerates, people introduce second-order necessary conditions (SONCs for short) for optimal controls. SONCs are well studied for optimal controls of systems described by ODEs and partial differential equations (PDEs for short). Some results for SONC of optimal controls for systems governed by stochastic differential equations (SDEs for short) are also obtained. However, nothing is known about the SONC for optimal controls of systems described by stochastic (infinite dimensional) evolution equations (SEEs for short). This paper aims to give a solution to this difficult problem, under some mild conditions.

In this paper, we study the null controllability of linear heat and wave equations with spatial nonlocal integral terms. Under some analyticity assumptions on the corresponding kernel, we show that the equations are controllable. We employ compactness-uniqueness arguments in a suitable functional setting, an argument that is harder to apply for heat equations because of its very strong time irreversibility. Some possible extensions and open problems concerning other nonlocal systems are presented, E.F.C. was partially supported by MINECO (Spain) grant MTM2013-41286-P. Q.L. was supported by the NSF of China under grant 11471231, the Fundamental Research Funds for the Central Universities in China under grant 2015SCU04A02, and MICINN (Spain) grant MTM2011-29306-C02-00. E.Z. was partially supported by the advanced grant NUMERIWAVES/FP7-246775 of the European Research Council Executive Agency, FA9550-14-1-0214 of EOARD-AFOSR, FA9550-15-1-002 of AFOSR, MTM2011-29306 and MTM2014-52347 grants of MINECO, and a Humboldt Award at the University of Erlangen-Nurnberg

In [Yong 2004], it was proved that as long as the integrand has certain properties, the corresponding It\^o integral can be written as a (parameterized) Lebesgue integral (or a Bochner integral). In this paper, we show that such a question can be answered in a more positive and refined way. To do this, we need to characterize the dual of the Banach space of some vector-valued stochastic processes having different integrability with respect to the time variable and the probability measure. The later can be regarded as a variant of the classical Riesz Representation Theorem, and therefore it will be useful in studying other problems. Some remarkable consequences are presented as well, including a reasonable definition of exact controllability for stochastic differential equations and a condition which implies a Black�Scholes market to be complete.

This paper presents a time optimal control problem with control constraints of the rectangular type for internally controlled heat equations. An existence result of time optimal controls for such a problem is established. The rectangular type of control constraints originates from the study of time optimal control problems for ordinary differential equations. In the finite dimensional case, there is no difference between such problems with control constraints of the rectangular type and those of the ball type, from the perspective of the study on the existence of optimal controls. Interestingly, in the infinite dimensional case, the problem with control constraints of the rectangular type differs essentially from that with control constraints of the ball type. For infinite dimensional systems, the existence for time optimal controls with constraints of the ball type has already been discussed in the literature, while the study of the rectangular type has not been touched upon as far as we know.

In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of ℝn, with a controller ω to be any given nonempty open subset of ω. The problem is reduced to a new controllability property for this equation, i.e. the null controllability of the system at any given time T > 0 when the control is restricted to be active in ω × E, where E is any given subset of [0, T] with positive (Legesgue) measure. The desired controllability result is established by means of a sharp observability estimate on the eigenfunctions of the Dirichlet Laplacian due to Lebeau and Robbiano, and a delicate result in the measure theory due to Lions.

This paper is concerned with a state observation problem for the general time reversible system, i.e., to recover the initial state of the given system from suitable observation data. An abstract unified framework is presented. By means of an iterative time reversal technique, we derive two asymptotic formulae of reconstructing the initial state approximately with explicit convergence rates. General results are then applied to the state observation problems for the Schrodinger equation and the wave equation.

International audience; This paper is devoted to the study of the local rapid exponential stabilization problem for a controlled Kuramoto–Sivashinsky equation on a bounded interval. We build a feedback control law to force the solution of the closed-loop system to decay exponentially to zero with arbitrarily prescribed decay rates, provided that the initial datum is small enough. Our approach uses a method we introduced for the rapid stabilization of a Korteweg–de Vries equation. It relies on the construction of a suitable integral transform and can be applied to many other equations.

The main purpose of this paper is to improve our transposition method to solve both vector-valued and operator-valued backward stochastic evolution equations with a general filtration. As its application, we obtain a general Pontryagin-type maximum principle for optimal controls of stochastic evolution equations in infinite dimensions. In particular, we drop the technical assumption appeared in [12, Theorem 9.1]. We also establish a Pontryagin-type maximum principle for a stochastic linear quadratic problems.

This paper is devoted to the study of the rapid exponential stabilization problem for a controlled Korteweg-de Vries equation on a bounded interval with homogeneous Dirichlet boundary conditions and Neumann boundary control at the right endpoint of the interval. For every noncritical length, we build a feedback control law to force the solution of the closed-loop system to decay exponentially to zero with arbitrarily prescribed decay rates, provided that the initial datum is small enough. Our approach relies on the construction of a suitable integral transform., Comment: 45 pages

In this chapter, we derive a general Pontryagin-type stochastic maximum principle for optimal controls with a possibly nonconvex control domain.

In this chapter, we prove a uniqueness result for transposition solutions to the operator-valued backward stochastic evolution Eq. (1.10) and a well-posedness result for transposition solutions to this equation for the special case that both the final datum and the nonhomogeneous term are valued in the Hilbert space of Hilbert-Schmidt operators.

We analyze the null controllability for heat equations in the presence of switching controls. The switching pattern is a priori unknown so that the control has to be designed in a robust manner, based only on the past dynamics, so to fulfill the final control requirement, regardless of what the future dynamics is. We prove that such a robust control strategy actually exists when the switching controllers are located on two non trivial open subsets of the domain where the heat process evolves. Our strategy to construct these robust controls is based on earlier works by Lebeau and Robbiano on the null controllability of the heat equation. It is relevant to emphasize that our result is specific to the heat equation as an extension of a property of finite-dimensional systems that we fully characterize but that it may not hold for wave-like equations.

In this paper, we establish a boundary observability estimate for stochastic Schrödinger equations by means of the global Carleman estimate. Our Carleman estimate is based on a new fundamental identity for a stochastic Schrödinger-like operator. Applications to the state observation problem for semilinear stochastic Schrödinger equations and the unique continuation problem for stochastic Schrödinger equations are also addressed.

This paper is addressed to studying the exact controllability for stochastic transport equations by two controls: one is a boundary control imposed on the drift term and the other is an internal control imposed on the diffusion term. By means of the duality argument, this controllability problem can be reduced to an observability problem for backward stochastic transport equations, and the desired observability estimate is obtained by a new global Carleman estimate. Also, we present some results about the lack of exact controllability, which show that the action of two controls is necessary. To some extent, this indicates that the controllability problems for stochastic PDEs differ from their deterministic counterpart.

This paper is addressed to the boundary unique continuation property for forward stochastic parabolic equations, that is, to determine the value of the solution by virtue of the observation on an arbitrary open subset of the boundary. By means of a global Carleman estimate, we establish a quantitative version of this property.

The authors establish the null controllability for some systems coupled by two backward stochastic heat equations. The desired controllability result is obtained by means of proving a suitable observability estimate for the dual system of the controlled system.

In this paper, we establish a global Carleman estimate for stochastic parabolic equations. Based on this estimate, we solve two inverse problems for stochastic parabolic equations. One is concerned with a determination problem of the history of a stochastic heat process through the observation at the final time $T$, for which we obtain a conditional stability estimate. The other is an inverse source problem with observation on the lateral boundary. We derive the uniqueness of the source., 18 pages

This paper is addressed to the well-posedness of some linear and semilinear backward stochastic differential equations with general filtration, without using the Martingale Representation Theorem. The point of our approach is to introduce a new notion of solution, i.e., the transposition solution, which coincides with the usual strong solution when the filtration is natural but it is more flexible for the general filtration than the existing notion of solutions. A comparison theorem for transposition solutions is also presented., 21 pages

This paper is devoted to a study of the boundary and internal state observation problems for stochastic hyperbolic equations. For this, we derive a boundary and an internal observability inequality for stochastic hyperbolic equations with nonsmooth lower order terms. The required inequalities are obtained by the global Carleman estimate for stochastic hyperbolic equations. By these inequalities, we get stability estimates for the state observation problems. As a consequence, we also establish a unique continuation property for stochastic hyperbolic equations.