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In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly reduce the computational load and storage requirements associated with matrix inversion without losing accuracy. To demonstrate the versatility and applicability of our method, we apply it to address two crucial uncertainty quantification problems: stochastic elliptic equations and optimal control problems governed by stochastic elliptic PDE constraints. Based on varying dimension reduction ratios, our algorithm exhibits the capability to yield a high-precision numerical solution for stochastic partial differential equations, or provides a rough representation of the exact solutions as a pre-processing phase. Meanwhile, our algorithm for solving stochastic optimal control problems allows a diverse range of gradient-based unconstrained optimization methods, rendering it particularly appealing for computationally intensive large-scale problems. Numerical experiments are conducted and the results provide strong validation of the feasibility and effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

Two key challenges in optimal control include efficiently solving high-dimensional problems and handling optimal control problems with state-dependent running costs. In this paper, we consider a class of optimal control problems whose running costs consist of a quadratic on the control variable and a convex, non-negative, piecewise affine function on the state variable. We provide the analytical solution for this class of optimal control problems as well as a Hopf-type representation formula for the corresponding Hamilton-Jacobi partial differential equations. Finally, we propose efficient numerical algorithms based on our Hopf-type representation formula, convex optimization algorithms, and min-plus techniques. We present several high-dimensional numerical examples, which demonstrate that our algorithms overcome the curse of dimensionality. We also describe a field-programmable gate array (FPGA) implementation of our numerical solver whose latency scales linearly in the spatial dimension and that achieves approximately a 40 times speedup compared to a parallelized central processing unit (CPU) implementation. Thus, our numerical results demonstrate the promising performance boosts that FPGAs are able to achieve over CPUs. As such, our proposed methods have the potential to serve as a building block for solving more complicated high-dimensional optimal control problems in real-time. [ABSTRACT FROM AUTHOR]

In this paper, elliptic control problems with the integral constraint on the gradient of the state and the box constraint on the control are considered. The optimality conditions for the problem are proved. To numerically solve the problem, a finite element duality-based inexact majorized accelerated block coordinate descent (FE-dABCD) algorithm is proposed. Specifically, both the state and the control are discretized by piecewise linear functions. An inexact majorized ABCD algorithm is employed to solve the discretized problem via its dual, which is a multi-block unconstrained convex optimization problem, but the primal variables are also generated in each iteration. Thanks to the inexactness of the FE-dABCD algorithm, the subproblems at each iteration are allowed to be solved inexactly. For the smooth subproblem, we use the preconditioned generalized minimal residual (GMRES) method to solve it. For the two nonsmooth subproblems, one of them has a closed form solution through introducing an appropriate proximal term, and another one is solved by the line search Newton's method. Based on these efficient strategies, we prove that our proposed FE-dABCD algorithm enjoys O (1 k 2 ) iteration complexity. Moreover, to make the algorithm more efficient and further reduce its computation cost, based on the mesh-independence of ABCD method, we propose an FE-dABCD algorithm with a warm-start strategy (wFE-dABCD). Some numerical experiments are done and the numerical results show the efficiency of the FE-dABCD algorithm and wFE-dABCD algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an optimal control problem (OCP) for three-dimensional (3D) unsteady partial differential equations with convection term (UPDECT), extensively applied to the solidification process of billets. Firstly, we introduce the mathematical model of OCP and adopt a life cycle model method (LCMM) to numerically solve the 3D UPDECT. Secondly, the Lipschitz continuity of the cost function gradient is proved using the adjoint approach as a numerical method for designing optimal control systems. The cost function gradient of OCP for 3D UPDECT is analyzed based on a Hamiltonian costate system. Thirdly, we propose a modified trinomial conjugate gradient method (MTCGM) to solve the OCP for 3D UPDECT and prove the global convergence of MTCGM. Finally, the MTCGM is applied to the experimental simulation of continuous casting. According to experiment results, the optimizer based on MTCGM can significantly accelerate the convergence speed and reduce the iterative numbers, and the MTCGM provides a more stable temperature control effect. • An optimal control problem (OCP) for three-dimensional unsteady partial differential equation is studied. • The Lipschitz continuity of cost function gradient in the OCP is proved, and the Lipschitz constant is obtained. • The expression of gradient calculation of the OCP is analyzed based on Hamiltonian costate system. • A modified trinomial conjugate gradient method (MTCGM), and the global convergence of the MTCGM is proved. • The effectiveness of MTCGM is verified by experimental simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal control problem governed by linear parabolic differential equations with memory. Under the assumption that the corresponding linear parabolic differential equation without memory term is approximately controllable, it is shown that the set of approximate controls is nonempty. The problem is first viewed as a constrained optimal control problem, and then it is approximated by an unconstrained problem with a suitable penalty function. The optimal pair of the constrained problem is obtained as the limit of the optimal pair sequence of the unconstrained problem. The result is proved by using the theory of strongly continuous semigroups and the Banach fixed point theorem. The approximation theorems, which guarantee the convergence of the numerical scheme to the optimal pair sequence, are also proved. Finally, we present an algorithm to compute an optimal control with a numerical example. [ABSTRACT FROM AUTHOR]

The paper presents a numerical procedure for solving a class of optimal control problems for heterogeneous systems. The latter are described by parameterized systems of ordinary differential equations, coupled by integrals along the parameter space. Such problems arise in economics, demography, epidemiology, management of biological resources, etc. The numerical procedure includes discretization and a gradient projection method for solving the resulting discrete problem. A main point of the paper is the performed error analysis, which is based on the property of metric regularity of the system of necessary optimality conditions associated with the considered problem. [ABSTRACT FROM AUTHOR]

In this paper, finite point method (FPM) and meshless weighted least squares (MWLS) method are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The FPM scheme uses shape function constructed by moving least square (MLS) approximation to discretize the equations, while the MWLS scheme employs both MLS approximation and penalty terms to solve the same problem. Error estimates for the FPM scheme are presented and numerical results are provided to examine the impact of parameters and validate the efficiency of the proposed schemes. The extended model (Navier–Stokes equations) shows the ability of our algorithm to handle complex problems. Our explorative work shows the flexibility and great potential of the meshless methods in optimal control problems. [ABSTRACT FROM AUTHOR]

We consider a Markov chain approximation scheme for utility maximization problems in continuous time, which uses, in turn, a piecewise constant policy approximation, Euler–Maruyama time stepping, and a Gauß-Hermite approximation of the Gaußian increments. The error estimates previously derived in Picarelli and Reisinger (2018) are asymmetric between lower and upper bounds due to the control approximation and improve on known results in the literature in the lower case only. In the present paper, we use duality results to obtain a posteriori upper error bounds which are empirically of the same order as the lower bounds. The theoretical results are confirmed by our numerical tests. [ABSTRACT FROM AUTHOR]

In this paper, we compare three model order reduction methods: the proper orthogonal decomposition (POD), discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) for the optimal control of the convective FitzHugh–Nagumo (FHN) equations. The convective FHN equations consist of the semi-linear activator and the linear inhibitor equations, modeling blood coagulation in moving excitable media. The semilinear activator equation leads to a non-convex optimal control problem (OCP). The most commonly used method in reduced optimal control is POD. We use DEIM and DMD to approximate efficiently the nonlinear terms in reduced order models. We compare the accuracy and computational times of three reduced-order optimal control solutions with the full order discontinuous Galerkin finite element solution of the convection dominated FHN equations with terminal controls. Numerical results show that POD is the most accurate whereas POD–DMD is the fastest. [ABSTRACT FROM AUTHOR]

Abstract: This paper develops the data-assimilation procedure in order to allow for the assimilation of measurements of currents and free-surface elevations into an unsteady flow solution governed by the free-surface barotropic Navier-Stokes equations. The flow is considered in a 2D vertical section in which horizontal and vertical components of velocity are represented as well as the elevation of the free surface. Since a possible application is to the construction of a coastal (limited area) circulation model, the open boundary control problem is the main scope of the paper. The assimilation algorithm is built on the limited memory quasi-Newton LBFGS method guided by the adjoint sensitivities. The analytical step search, which is based on the solution of the tangent linear model, is used. We process the gradients to regularize the solution. In numerical experiments we consider different wave patterns with a purpose to specify a set of incomplete measurements, which could be sufficient for boundary-control identification. As a result of these experiments we formulate some important practical conclusions. [Copyright &y& Elsevier]

In this paper, employing a fixed point-collocation method, we solve an optimal control problem for a model of tumor growth with drug application. This model is a free boundary problem and consists of five time-dependent partial differential equations including three different first-order hyperbolic equations describing the evolution of cells and two second-order parabolic equations describing the diffusion of nutrient and drug concentration. In the mentioned optimal control problem, the concentration of nutrient and drug is controlled using some control variables in order to destroy the tumor cells. In this study, applying the fixed point method, we construct a sequence converging to the solution of the optimal control problem. In each step of the fixed point iteration, the problem changes to a linear one and the parabolic equations are solved using the collocation method. The stability of the method is also proved. Some examples are considered to illustrate the efficiency of method. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate the Legendre–Galerkin spectral approximation of elliptic optimal control problems with integral state and control constraints. Thanks to the appropriate base functions of the discrete spaces, the discrete system is with sparse coefficient matrices. We first present the optimality conditions of the control system. Then a priori and a posteriori error estimates both in H 1 and L 2 norms are derived. Some numerical tests indicate that the spectral accuracy can be achieved, and the proposed method is competitive for solving control problems. [ABSTRACT FROM AUTHOR]

Abstract: A two-dimensional (2D) concentration field may be reconstructed using inverse advection–diffusion–reaction based on pointwise sensor outputs. The studied reconstruction process consists in defining fictitious boundaries and identifying the optimal concentrations on these boundaries that minimize the data misfit. The minimization is performed by means of the adjoint method. The reconstruction process is observed to fail for some sets of sensor placements. Therefore, in this paper, we aim to develop a low computational cost framework that can be used to identify badly placed sensors and to reveal potentially optimal sensor placement. From this framework, we have derived a sensitivity tool that can predict, for a given set of sensors, which boundary concentrations are easy or difficult to identify using the inverse method. This tool only requires the resolution of an adjoint problem. It is compared with the “sensitivity-to-observation” method in a 2D water pipe junction with 2⋅103 Reynolds number and 103 Peclet number. The compared methods do produce similar quantitative and qualitative results. [Copyright &y& Elsevier]

Abstract: The present paper discusses chaos, estimation and optimal control of the habitat destruction model with unknown parameters. The linear stability analysis of the steady states of the model will be discussed. Further, the chaotic behavior of this system will be investigated. The dynamic estimators of the unknown parameters and their updating rules are derived. Using Pontryagin principle, the optimal control inputs are derived with respect to a selected measure. Finally, a numerical simulation study for various parameters and different initial densities is presented. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the state constrained optimal control problems of a fractional diffusion equation in a bounded domain. The fractional time derivative is considered in the Riemann–Liouville sense. Under a Slater type condition we prove the existence a Lagrange multiplier and a decoupled optimality system. [Copyright &y& Elsevier]

Abstract: This paper is mainly concerned with a class of optimal control problems of a system governed by the nonlinear integro-differential equation on time scales. Using the contraction mapping principle, the existence and uniqueness of a weak solution of the nonlinear integro-differential equation on time scales are proved. Discussing -strong–weak lower semicontinuity of integral functional on time scales, we give sufficient conditions for the existence of optimal control. By the virtue of the integration by parts formula on time scales, the necessary conditions of optimality are derived. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we apply the classical control theory to a fractional diffusion equation in a bounded domain. The fractional time derivative is considered in a Riemann–Liouville sense. We first study the existence and the uniqueness of the solution of the fractional diffusion equation in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. Interpreting the Euler–Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. [ABSTRACT FROM AUTHOR]

Abstract: Recent advances in the power system handling capabilities of static switches have made the use of the voltage source inverter (VSI) feasible at both transmission and distribution levels. This paper is focused on comparison between optimal control strategies applied to a system with unified power flow controller, shunt converter and series converter. The control strategies were tested using a power system distribution model. The presented results show that when the system is with one converter, there will be only two control inputs in the system, making it possible to track only two outputs in the system. If the objective is to maintain capacitor voltage constant, it is possible to track one output of the system for the capacitor voltage to be maintained constant. When the system is with two converters, then there will be four control inputs in system, and so it is possible to track four outputs in the system. The series converter can eliminate the voltage flicker that exists in the same branch the series converter is located in, while the shunt converter can eliminate the current harmonics. The UPFC can remove both the current harmonics and voltage flicker from the system. [ABSTRACT FROM AUTHOR]

Abstract: Recent advances in the power system handling capabilities of static switches have made the use of the voltage source inverter (VSI) feasible at both transmission and distribution levels. Most of the control schemes introduced were designed either for eliminating current harmonics or eliminating voltage flickers or for load flow control. They are mostly based on the PI controller. None of these control schemes were based on a systematic optimal control law that can control the converters to achieve all purposes at the same time, load flow control, voltage flicker elimination and also current harmonics elimination. This paper presents three novel control strategies: optimal control with tracking according to desired steady-state behaviour, optimal control with linear quadratic tracker design, and optimal control with tracking with PI controllers. The strategies were tested using a power system distribution model. The presented results show that all three control strategies improve the performance of the load flow and eliminate the current harmonics and voltage flicker. The controller design based on the desired steady-state behaviour is superior compared to other way of designs as the low value of the weighting matrix Q is satisfactory to obtain the desired tracking. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an innovative procedure to solve the Connection Admission Control Problem for a telecommunication network. Here, this important problem in the context of Communication Theory and Network Dynamics is dealt with by imbedding it in the framework of System and Control Theory. Highlights of the procedure are technology independence, coordinated and coherent decoupling, optimality and feedback properties, stochastic dynamic control. [Copyright &y& Elsevier]

Abstract: In this paper, a class of second-order nonlinear impulsive integro-differential equations of mixed type whose principle is time-varying generating operators with unbounded perturbation on Banach spaces is considered. Discussing the perturbation of time-varying operator matrix and constructing corresponding the evolution system generated by operator matrix, we introduce the reasonable mild solution of second-order nonlinear impulsive integro-differential equations of mixed type and prove the existence of mild solutions. The existence of optimal controls for a Lagrange problem of systems governed by the second-order nonlinear impulsive integro-equations of mixed type is also presented. An example is given for demonstration. [Copyright &y& Elsevier]

Abstract: In this paper, optimal control for stochastic nonlinear singular system with quadratic performance is obtained using neural networks. The goal is to provide optimal control with reduced calculus effort by comparing the solutions of the matrix Riccati differential equation (MRDE) obtained from the well-known traditional Runge–Kutta (RK) method and nontraditional neural network method. To obtain the optimal control, the solution of MRDE is computed by feedforward neural network (FFNN). The accuracy of the solution of the neural network approach to the problem is qualitatively better. The advantage of the proposed approach is that, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to construct an unconstrained optimal control problem by using a least-squares approach for the constrained distributed optimal control problem associated with incompressible Stokes equations. The constrained equations are reformulated to the equivalent first-order system by introducing vorticity, and then the least-squares functional corresponding to the system is enforced via a penalty term to the objective functional. The existence of a solution of the unconstrained optimal control problem is proved, and the convergence of this solution to that of unpenalized one is demonstrated as the penalty parameter tends to zero. Finite element approximations with error estimates are studied, and the relevant computational experiments are presented. [Copyright &y& Elsevier]

This paper considers an infinite-horizon optimal tracking control problem for a class of large-scale interconnected systems with state time-delays. By using the successive approximation approach, two iteration sequences of vector differential equations are constructed. Meanwhile the large-scale interconnected system is decomposed into finite decoupled subsystems. The existence and uniqueness of the optimal solution is proved, as well as the convergence of the solution sequence. By finite iterations of the solution sequence, a suboptimal tracking control law is obtained. A reduced-order reference input observer is designed to make the feedforward term of the optimal tracking control law physically realizable. A numerical example shows that the presented algorithm is effective and easy to implement. [Copyright &y& Elsevier]

Abstract: This paper presents a numerical application of the optimal control considering time- delay system to the flood control with drainage basin. The optimal control theory is used to obtain the control values which satisfies the state equation and minimizes performance function. Feature of this study is application of time-delay system to control problems. Two kinds of control values which has time difference can be introduced. Those are quantity of water intake and release in drainage basin. These control values are correlated with each other, which can be modeled by the time-delay system. The Sakawa-Shindo method is employed for the minimization algorithm. To calculate the water flow phenomenon, the shallow water equation is employed. The Clank-Nicolson method is applied to temporal discretization and the stabilized bubble function method is applied to spatial discretization. As the numerical model, the flood flow in the Tsurumi river which locates in Yokohama city is represented. [Copyright &y& Elsevier]

Abstract: This paper develops the adjoint sensitivities to the free-surface barotropic Navier- Stokes equations in order to allow for the assimilation of measurements of currents and free-surface elevations into an unsteady flow solution by open-boundary control. To calculate a variation in a surface variable, a mapping is used in the vertical to shift the problem into a fixed domain. A variation is evaluated in the transformed space from the Jacobian matrix of the mapping. This variation is then mapped back into the original space where it completes a tangent linear model. The adjoint equations are derived using the scalar product formulas redefined for a domain with variable bounds. The method is demonstrated by application to an unsteady fluid flow in a one-dimensional open channel in which horizontal and vertical components of velocity are represented as well as the elevation of the free surface (a 2D vertical section model). This requires the proper treatment of open boundaries in both the forward and adjoint models. A particular application is to the construction of a fully three-dimensional coastal ocean model that allows assimilation of tidal elevation and current data. However, the results are general and can be applied in a wider context. [Copyright &y& Elsevier]

In this paper, we consider the optimal feedback control problems of a system governed by strongly nonlinear impulsive differential equations which contains monotone operators and nonlinear nonmonotone perturbations. Based on the existence of feasible pairs, an existence result of optimal control pairs for the Lagrange problem is presented. [Copyright &y& Elsevier]

A shape design model that reduces the amount of noise radiated from aircraft turbofan engines is studied in this paper. The model is formulated as shape control of the Helmholtz equation with radiation boundary conditions on part of the boundary and incoming waves specified as the source. Existence of optimal shape is proved to show that the model is appropriately established. A numerical experiment is conducted to demonstrate the efficiency of the model. [Copyright &y& Elsevier]

In this paper, we present sufficient smoothness results on the solution of a nonlinear shallow water problem in order to obtain an existence result for the associated adjoint problem. We introduce a necessary but not sufficient existence condition for the functional minimum. The control is made on the velocity boundary condition, in considering observations at isolated points. Moreover, we give a set of numerical results obtained in a real situation. [Copyright &y& Elsevier]