Boundary Control of Parabolic-hyperbolic Systems with Time Delays

Keywords: boundary control, parabolic-hyperbolic system, integral time delays, Neumann problem, quadratic programming method, Hilbert space In this paper, we consider optimal boundary control problems for parabolic-hyperbolic systems in which time delays appear in the integral form in the state equations. The right-hand side of the state equation and the initial and boundary conditions are not continuous functions usually, but they are measurable functions belonging into L 2 or L ∞ spaces. Therefore we shall look the solution of the time delay parabolic-hyperbolic equation in some Sobolev spaces (Lions and Magenes, 1972). Sufficient conditions for the existence of a unique solution of the time delay parabolic – hyperbolic equation with the Neumann boundary condition are proved – Theorem 1. Consequently, we formulate the optimal control problem. The performance functional has the quadratic form. The time horizon is fixed. Finally, we impose some constraints on the boundary control. The necessary and sufficient conditions of optimality with the quadratic performance functional and constrained control are derived for the Neumann problem-Theorem 2. We must notice that the conditions of optimality derived above (Theorem 2) allow us to obtain an analytical formula for the optimal control in particular cases only (e.g. there are no constraints on the controls). This results from the following: the determining of the adjoint variable p(v 0) in the maximum condition from the adjoint equation is possible if and only if we know the state variable y 0 which corresponds to the control v 0. These mutual connections make the practical use of the derived optimization formulae difficult. Therefore we resign from the exact determining of the optimal control and we use approximation methods (Kowalewski, 2001). As an example, a quadratic programming method in a Hilbert space, which can be used in solving certain optimization problems for time delay parabolic-hyperbolic systems is also presented. The results presented in the paper can be treated as an extension of the results obtained by Lions (1971) onto the case of additional time delays appearing in the integral form in the state equations. In this paper we have considered the optimal time delay parabolic-hyperbolic systems with