Optimal Control and Feedback Design of State-Constrained Parabolic Systems in Uncertainty Conditions

The paper concerns minimax control problems for linear multidimensional parabolic systems with distributed uncertain perturbations and control functions acting in the Dirichlet boundary conditions. The underlying parabolic control system is functioning under hard/pointwise constraints on control and state variables. The main goal is to design a feedback control regulator that ensures the required state performance and robust stability under any feasible perturbations and minimize an energy-type functional under the worst perturbations from the given area. We develop a constructive approach to the minimax control design of constrained parabolic systems that is based on certain characteristic features of the parabolic dynamics including the transient monotonicity with respect to both controls and perturbations and the turnpike asymptotic behavior on the infinite horizon. In this way, solving a number of associated open-loop control and optimization problems, we justify an easily implementable three-positional suboptimal structure of the feedback boundary regulator and compute its optimal parameters, ensuring thus the required state performance and robust stability of the closed-loop, highly nonlinear parabolic control system on the infinite horizon

Asymmetric Games for Convolution Systems with Applications to Feedback Control of Constrained Parabolic Equations

The paper is devoted to the study of some classes of feedback control problems for linear parabolic equations subject to hard/pointwise constraints on both Dirichlet boundary controls and state dynamic/output functions in the presence of uncertain perturbations within given regions. The underlying problem under consideration, originally motivated by automatic control of the groundwater regime in irrigation networks, is formalized as a minimax problemof optimal control, where the control strategy is sought as a feedback law. Problems of this type are among the most important in control theory and applications - while most challenging and difficult. Based on the Maximum Principle for parabolic equations and on the time convolution structure, we reformulate the problems under consideration as certain asymmetric games, which become the main object of our study in this paper. We establish some simple conditions for the existence of winning and losing strategies for the game players, which then allow us to clarify controllability issues in the feedback control problem for such constrained parabolic systems

Minimax Control of Constrained Parabolic Systems

In this paper we formulate and study a minimax control problem for a class of parabolic systems with controlled Dirichlet boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We prove an existence theorem for minimax solutions and develop effective penalized procedures to approximate state constraints. Based on a careful variational analysis, we establish convergence results and optimality conditions for approximating problems that allow us to characterize suboptimal solutions to the original minimax problem with hard constraints. Then passing to the limit in approximations, we prove necessary optimality conditions for the minimax problem considered under proper constraint qualification conditions

Robust Control of Constrained Parabolic Systems with Neumann Boundary Conditions

This paper presents recent results by the authors on minimax robust control design of parabolic systems with uncertain perturbations under pointwise state and control constraints. The design procedure involves multi-step approximations and essentially employs monotonicity properties of the parabolic dynamics as well as its asymptotics on the infinite horizon. The results obtained justify a suboptimal three-positional structure of feedback controllers in the Neumann boundary conditions and provide calculations of their optimal parameters to ensure the required state performance and stability under any admissible perturbations. The problem under consideration was originally motivated by control design in water resources but certainly admits a much broader spectrum of applications

Suboptimal Feedback Control Design of Constrained Parabolic Systems in Uncertainty Conditions

The paper concerns minimax control problems forlinear multidimensional parabolic systems with distributed uncertain perturbations and control functions acting in the Dirichlet boundary conditions. The underlying parabolic control system is functioning under hard/pointwise constraints on control and state variables. The main goal is to design a feedback control regulator that ensures the required state performance and robust stability under any feasible perturbations and minimize an energy-type functional under the worst perturbations from the given area. We develop an efficient approach to the minimax control design of constrained parabolic systems that is based on certain characteristic features of the parabolic dynamics including the transient monotonicity with respect to both controls and perturbations and the turnpike asymptotic behavior on the infinite horizon. In this way, solving a number of associated open-loop control and approximation problems, we justify an easily implemented suboptimal structure of the feedback boundary regulator and compute its optimal parameters ensuring the required state performance and robust stability of the closed-loop, highly nonlinear parabolic control system on the infinite horizon

Optimization and Feedback Design of State-Constrained Parabolic Systems

The paper is devoted to optimal control and feedback design of stateconstrained parabolic systems in uncertainty conditions. Problems of this type are among the most challenging and difficult in dynamic optimization for any kind of dynamical systems. We pay the main attention to considering linear multidimensional parabolic\u27systems with Dirichlet boundary controls and pointwise state constraints, while the methods developed in this study are applicable to other kinds of boundary controls and dynamical systems of the parabolic type. The feedback design problem is formulated in the minimax sense to ensure stabilization of transients within the prescribed diapason and robust stability of the closed-loop control system under all feasible perturbations with minimizing an integral cost functional in the worst perturbation case. Exploiting certain fundamental properties of the parabolic dynamics, we determine the worst perturbations in the minimax control problem and efficiently solve the associated optimal control problems for approximating ODE and the original PDE systems with pointwise state constraints: In this way, using the transient monotonicity and turnpike asymptotic properties of the underlying parabolic dynamics on the infinite horizon, we compute optimal (in the minimax sense) parameters of the easily implemented while rigorously justified three-positional suboptimal structure of the feedback boundary controls that ensure robust stability of the closed-loop and highly nonlinear parabolic control system under consideration

Suboptimal Minimax Design of Constrained Parabolic Systems with Mixed Boundary Control

The paper concerns minimax control problems for linear multidimensional parabolic systems with distributed uncertain perturbations and control functions acting in mixed (Robin) boundary conditions. The main goal is to design a feedback control regulator that ensures the required state performance and robust stability under any feasible perturbations and minimize an energy-type functional under the worst perturbations from the given area. We design and justify an easily implemented suboptimal structure of the feedback boundary regulator and compute its optimal parameters ensuring the required state performance and robust stability of the nonlinear closed-loop control system on the infinite horizon

Optimal Moving Sensors for Parabolic Systems

This paper continues the investigations in SDS on observability issues motivated by environmental monitoring and related problems. Here the author introduces a specific class of scanning sensors that ensure solvability of the problem and can further lead to numerically robust techniques

Robust filtering for a class of nonlinear stochastic systems with probability constraints

This paper is concerned with the probability-constrained filtering problem for a class of time-varying nonlinear stochastic systems with estimation error variance constraint. The stochastic nonlinearity considered is quite general that is capable of describing several well-studied stochastic nonlinear systems. The second-order statistics of the noise sequence are unknown but belong to certain known convex set. The purpose of this paper is to design a filter guaranteeing a minimized upper-bound on the estimation error variance. The existence condition for the desired filter is established, in terms of the feasibility of a set of difference Riccati-like equations, which can be solved forward in time. Then, under the probability constraints, a minimax estimation problem is proposed for determining the suboptimal filter structure that minimizes the worst-case performance on the estimation error variance with respect to the uncertain second-order statistics. Finally, a numerical example is presented to show the effectiveness and applicability of the proposed method