Solvability Of Optimal Control Problem Research Articles

Optimal Control of Quasi-Stationary Equations of Complex Heat Transfer with Reflection and Refraction Conditions

The paper considers a class of optimal control problems for a nonlinear parabolic-elliptic system simulating radiative heat transfer with Fresnel matching conditions on surfaces of discontinuity of the refractive index. New estimates for the solution of the initial-boundary value problem are obtained, on the basis of which the solvability of optimal control problems is proved. Non-degenerate first-order optimality conditions are derived. The results are examplified by control problems with final, boundary, and distributed observations.

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.

Decentralized linear–quadratic–Gaussian control of networked control systems with asymmetric information

AbstractThe decentralized linear–quadratic–Gaussian (LQG) control problem for networked control systems (NCSs) with asymmetric information is investigated, where controller 1 shares its historical information with controller 2, and not vice versa. The asymmetry of the information structure leads to the coupling between controller 2 and estimator 1, and hence the classical separation principle fails. Through the assumption of linear control strategy, the coupling between controller 2 and estimator 1 (CCE) is decoupled, but the estimation gain is still coupled with the control gain. It is noted that the control gain conforms to the backward Riccati equation while estimation gain abides by the forward equation, which is computationally challenging. Applying the stochastic maximum principle, the solvability of the decentralized LQG control problem is reduced to that of corresponding forward and backward stochastic difference equations (FBSDEs). Further, necessary and sufficient conditions for the solvability of optimal control problem are presented by two Riccati equations, one of which is nonsymmetric. Moreover, a novel iterative forward method is proposed to calculate the coupled backward control gain and forward estimation gain.

Mixed optimal control for discrete-time stochastic systems with random coefficients

This paper is concerned with the problem of mixed optimal control for discrete-time systems with random coefficients where “mixed” means that there are two controllers: the deterministic controller and the random controller. Different from previous works, random coefficients and controllers with different information structures arise simultaneously in the optimal control problem, which lead to large difficulty. Using decoupling technique to solve the forward and backward stochastic difference equations, the necessary and sufficient conditions for the solvability of optimal control problem are given. An explicit expression of the optimal control strategies is derived based on a stochastic Riccati-type equation, which is new to the best of knowledge. Finally, a pension management problem is solved with the obtained theoretical results.

Optimal control and stabilization for NCSs with multiple memoryless channels and hold‐input compensation

This paper investigates the optimal control problem for networked control systems over multiple Markovian packet loss channels under hold-input compensation strategy. In this situation, the actuator receives the control signals from the controller via the individual channels. The system is transformed into a Markovian jump linear system by introducing a new Markovian jump variable. By giving a new maximum principle for such a system, a coupled difference Riccati-type equation is developed, which in turn gives necessary and sufficient conditions for the solvability of optimal control problems in the finite horizon. Furthermore, for the infinite horizon case, the Markovian jump linear system is mean square stabilizable if and only if a set of coupled algebraic Riccati-type equation has a particular solution. Finally, numerical simulations are presented to illustrate the control effects of the proposed optimal controllers.

Strong Solution and Optimal Control Problems for a Class of Fractional Linear Equations

In this paper, we examine the unique solvability (in the sense of strong solutions) of the Cauchy problem for a linear inhomogeneous equation in a Banach space solved with respect to the Caputo fractional derivative. We assume that the operator acting on the unknown function in the right-hand side of the equation generates an analytic resolving operator family for the corresponding homogeneous equation. We obtain a representation of a strong solution of the Cauchy problem and examine the solvability of optimal control problems with a convex, lower semicontinuous, lower bounded, coercive functional for the equation considered. The general results obtained are used to prove the existence of an optimal control in problems with specific functionals. Abstract results obtained for a control system described by an equation in a Banach space are illustrated by examples of optimal control problems for a fractional equation whose special cases are the subdiffusion equation and the diffusion wave equation.

Optimal control for discrete-time NCSs with input delay and Markovian packet losses: Hold-input case

This paper is concerned with the linear–quadratic optimal control problem for networked systems simultaneously with input delay and Markovian packet losses under hold-input compensation strategy, which is different from the literature. Necessary and sufficient conditions for the solvability of optimal control problem over a finite horizon are given by coupled difference Riccati-type equations. Moreover, the networked control system is mean-square stability if and only if coupled algebraic Riccati-type equations have a particular solution. Due to input delay and Markovian packet dropout, it leads to the failure of the separation principle, which is a fundamental obstacle. The key technique in this paper is to tackle forward and backward difference equations by decoupling.

Optimal control problems governed by 1-D Kobayashi–Warren–Carter type systems

This paper is devoted to the study of a class of optimal control problems governed by 1–D Kobayashi–Warren–Carter type systems, which are based on a phase-field model of grain boundary motion, proposed by [Kobayashi et al, Physica D, 140, 141–150, 2000]. The class consists of an optimal control problem for a physically realistic state-system of Kobayashi–Warren–Carter type, and its regularized approximating problems. The results of this paper are stated in three Main Theorems 1–3. The first Main Theorem 1 is concerned with the solvability and continuous dependence for the state-systems. Meanwhile, the second Main Theorem 2 is concerned with the solvability of optimal control problems, and some semi-continuous association in the class of our optimal control problems. Finally, in the third Main Theorem 3, we derive the first order necessary optimality conditions for optimal controls of the regularized approximating problems. By taking the approximating limit, we also derive the optimality conditions for the optimal controls for the physically realistic problem.

Linear quadratic regulation for discrete-time systems with input delay and colored multiplicative noise

This paper is concerned with the linear quadratic regulation (LQR) problem for discrete-time systems involving input delay and colored multiplicative noise. Due to the correlation of the adjoining state, this problem will be more complicated than the case of white noise. In this paper, the necessary and sufficient condition for the solvability of optimal control problem is presented by solving the forward and backward stochastic difference equations (FBSDEs). Moreover, the optimal LQR controller is given in terms of the coupled difference equations developed in this paper.

On the partial controllability of SDEs and the exact controllability of FBSDES

A notion of partial controllability (also can be called directional controllability or output controllability) is proposed for linear controlled (forward) stochastic differential equations (SDEs), which characterizes the ability of the state to reach some given random hyperplane. It generalizes the classical notion of exact controllability. For time-invariant system, checkable rank conditions ensuring SDEs’ partial controllability are provided. With some special setting, the partial controllability for SDEs is proved to be equivalent to the exact controllability for linear controlled forward-backward stochastic differential equations (FBSDEs). Moreover, we obtain some equivalent conclusions to partial controllability for SDEs or exact controllability for FBSDEs, including the validity of observability inequalities for the adjoint equations, the solvability of some optimal control problems, the solvability of norm optimal control problems, and the non-singularity of a random version of Gramian matrix.

Optimization of the shapes of obstacles in jet-separation flow

The model of an ideal incompressible fluid is used to study the solvability of optimal control problems for the shape of a nozzle which discharges free-boundary fluid flow with and without accounting for gravity (internal aerodynamics) and shape optimization problems for an obstacle with jet separation (external aerodynamics). The qualitative properties of such flows are studied.