Kaixia Zhang

Dirichlet Boundary Control of Parabolic Systems with Pointwise State Constraints

In this paper we study an optimal control problem for linear parabolic systems with pointwise state constraints and measurable controls acting in the Dirichlet boundary conditions. Using the framework of mild solutions to parabolic systems with nonregular dynamics, we prove a general existence theorem of optimal controls and derive necessary optimality conditions for the state-constrained problem under consideration. Our variational analysis is based on a well-posed penalization procedure to approximate state constraints and then to study a parametric family of approximating problems. The final result estab- lishes necessary optimality conditions for the original state-constrained problem by passing to the limit from approximating problems under a proper constraint qualification.

Approximation results for robust control of heat-diffusion equations

Mordukhovich (1990) and Mordukhovich and Zhang (1993) previously considered robust control problems for linear heat-diffusion equations with state constraints. Using multistep approximation procedures and taking into account special features of the problems, they obtained first-order characteristics and computed suboptimal parameters of three-positional boundary regulators. The main goal of our study is to find an appropriate feedback control law which ensures (minimax) optimality and robust stability properties of the closed-loop systems in the presence of uncertain perturbations. To the best of our knowledge, there are no results available in the general H/sub /spl infin//-control theory which could be directly applied to the distributed parameter systems considered under hard state-control constraints. High-order approximation results for the systems under consideration, which allow to obtain useful characteristics of suboptimal controls, are presented.

Feedback Control Design of Constrained Parabolic Systems in Uncertainty Conditions

The talk 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 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. The primary motivation for this study came from certain environmental models, in particular, those developed within the Dynamical System and Environmental Projects of the International Institute of Applied System Analysis (IIASA), Laxenburg, Austria. Department of Mathematics, Wayne State University, Detroit, Michigan 48202 (boris@math.wayne.edu). This research was partly supported by the USA National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-0451168.

Bang-bang principle for state-constrained parabolic systems with dirichlet boundary controls

In this paper we study a class of state-constrained boundary control problems for linear parabolic equations. Principal complications come from discontinuous and bounded Dirichlet boundary controllers which are motivated by applications to some robust control problems under uncertain perturbations. We develop an effective procedure to approximate state constraints and prove desirable convergences of such approximations. Using variational analysis, we establish necessary optimality conditions for approximation problems and then, passing to the limit under a proper constraint qualification, we prove necessary optimality conditions for the original state-constrained problem in the bang-bang principle form.<<ETX>>

Optimal control of state-constrained parabolic systems with nonregular boundary controllers

We study an optimal control problem of linear parabolic systems with pointwise state constraints and Dirichlet boundary controls. Our variational analysis is based on a well-posed penalization procedure. The obtained result establishes necessary optimality conditions for the original state-constrained problem by passing to the limit from approximating problems under a proper constraint qualification.

Feedback control of uncertain parabolic systems with hard constraints

This paper presents a method to design feedback control of constrained parabolic systems under uncertain perturbations with the aid of multistep approximations. The design procedure essentially employs monotonicity properties of parabolic dynamics and its asymptotic on the infinite horizon. The results obtained show that there is a suboptimal three-positional structure of feedback controllers through the Neumann boundary conditions. We also provide calculations of their optimal parameters to ensure the required state performance and stability under any admissible perturbations.

Suboptimality and stability analysis for feedback boundary control of heat-diffusion equations

We develop an effective approach to feedback control design of parabolic systems that takes into account specific properties of heat-diffusion and related dynamical processes. Some results obtained in this direction have been reported mostly for the case of Dirichlet boundary controls. Here we pay the main attention to justifying suboptimal characteristics of feedback controllers in Neumann boundary conditions and stability analysis of nonlinear closed-loop control systems obtained in this way. Based on the variational approach to the stability analysis and monotonicity properties of transients, we are able to reduce stability questions in the closed-loop nonlinear system to considering special optimal control problems with infinite horizon.

Minimax boundary control problem for parabolic systems with state constraints

We study a minimax Dirichlet boundary control problem with pointwise state constraints for a class of parabolic systems under unknown distributed perturbations. Our approach to optimality conditions is based on splitting the original minimax problem into two interrelated optimal control problems for distributed and boundary controllers with moving state constraints. Then we approximate the state-constrained linear systems by families of nonlinear systems with no state constraints by using effective penalization procedures. Based on the properties of mild solutions, we prove the variational convergence of approximations and obtain necessary optimality conditions for approximating solutions which ensure suboptimality conditions for the original minimax problems.

Robust optimal control for a class of distributed parameter system

In this paper, we study some optimal control and stabilization problems for N dimensional linear heat-diffusion equations with state constraints. The originial motivation for considering such problems came from the development of automatic control systems in irrigation networks which ensure an optimal groundwater regime under uncertain external perturbations; see Skaggs [8] and Mordukhovich [4] for more details. A core problem arising in these considerations is robust stabilization of the system dynamics by state-feedback controls. We study a class of distributed parameter systems where control appear in boundary conditions and have a bounded amplitude. The later creates essential difficulties in employing Hx-optimal control theory based on Riccati equations; see, e.g., Khargonekar et al. [3], Curtain et al. [2] and references therein.