© 1995 by Institute of Mathematics and its Applications
Control of distributed time delay systems with Dirichlet boundary conditions
Institute of Automatics, University of Mining and Metallurgy, al Mickiewicza 30, 30-059 Cracow, Poland
Various optimization problems associated with the optimal control of distributed parameter systems with time lags appearing in the boundary conditions have been studied recently by Wang (1975), Knowles (1978), Wong (1987), Kowalewski (1987a,b, 1988a,b,c, 1990a,b,c,d, 1991, 1993a,b,c,d, 1995) and Kowalewski & Duda (1992). In this paper optimal boundary control problems for distributed systems described by linear partial differential equations of parabolic and hyperbolic type in which constant time delays appear in the state equations are considered. Sufficient conditions for the existence of a unique solutions of such equations with the Dirichlet boundary conditions are proved. The performance functional has the quadratic form. The time horizon T is fixed. Finally, we impose some constraints on the control. Making use of Lions' scheme (Lions 1971) necessary and sufficient conditions of optimality for the Dirichlet problem with the quadratic performance functional and constrained control are derived. The flow chart of the algorithm, which can be used in the numerical solving of certain optimization problems for distributed parameter systems, is also presented.