Boundary Control of Parabolic Systems regularity of optimal solutions英文电子书.pdf

Appl. Math.Optim.4, 301-327 (1978) Applied Mathematics and Optimization Boundary Control of Parabolic Systems: Regularity of Optimal Solutions Irena Lasiecka Departmentof SystemScience,Universityof Californiaat Los Angeles,Los Angeles,California Communicatedby A. V. Balakrishnan Abstract. Boundary control problems for the linear, parabolic equations and a quadratic performance index are considered. The controls are allowed to be in the space L2[OT;L2(F)], where F is a boundary. Exploiting the semigroup approach, it is shown that optimal control belongs to Lz[OT;H1/Z(F)] and, as a consequence, optimal trajectory belongs to L I[OT;H 1(~2)].This result is obtained for two kinds of domains. The first are the domains with C ~-boundary and the second are the domains being the parallelepipeds. 1. Introduction We consider the linear quadratic optimal control problem for a class of parabolic equation with control on the boundary. Such problems have no analogue in ordinary differential equation and are unique to distributed parameter systems. Lions [10] has treated such a problem restricting his control to be in H 1/2(F) where F is C ~. From the application viewpoint this restriction seems to be somewhat artificial. Therefore, it is of interest to develop a theory in which the controls are allowed to be in L 2 on the boundary. In this case the first question is an acceptable formulation of such problems. Using semigroup theory, Balakrishnan [2] constructed the so-called mild solution for LE-boundary inputs. Further, developing analytical properties of the semigroup operator and