Time–optimal Boundary Control of a Parabolic System with Time Lags given in Integral Form

Various optimization problems associated with the optimal control of distributed parabolic systems with time delays appearing in boundary conditions were studied recently by Wang (1975), Knowles (1978), Kowalewski (1988; 1990a;1990b; 1998; 1999; 2001), Kowalewski and Duda (1992) and Kowalewski and Krakowiak (2000). In (Wang, 1975), optimal control problems for parabolic systems with Neumann boundary conditions involving constant time delays were considered. Such systems constitute, in a linear approximation, a universal mathematical model for many diffusion processes in which time-delayed feedback signals are introduced at the boundary of a system’s spatial domain. For example, in the area of plasma control, it is of interest to confine the plasma in a given bounded spatial domain Ω by introducing a finite electric potential barrier or a “magnetic mirror” surrounding Ω. For a collision-dominated plasma, its particle density is describable by a parabolic equation. Due to the particle inertia and finiteness of the electric potential barrier or the magnetic mirror field strength, the particle reflection at the domain boundary is not instantaneous. Consequently, the particle flux at the boundary of Ω at any time depends on the flux of particles which escaped earlier and reflected back into Ω at a later time. This leads to Neumann boundary conditions involving time delays. Necessary and sufficient conditions which optimal control must satisfy were derived. Estimates and a sufficient condition for the boundedness of solutions were obtained for parabolic systems with specified forms of feedback control.