| Abstract: | In this paper we consider the systems governed, by parabolioc equations[frac{{partial y}}{{partial t}} = sumlimits_{i,j = 1}^n {frac{partial }{{partial {x_i}}}} ({a_{ij}}(x,t)frac{{partial y}}{{partial {x_j}}}) - ay + f(x,t)]subject to the boundary control [frac{{partial y}}{{partial {nu _A}}}{|_sum } = u(x,t)] with the initial condition [y(x,0) = {y_0}(x)]We suppose that U is a compact set but may not be convex in [{H^{ - frac{1}{2}}}(Gamma )], Given [{y_1}( cdot ) in {L^2}(Omega )] and d>0, the time optimal control problem requiers to find the control[u( cdot ,t) in U] for steering the initial state {y_0}( cdot )] the final state [left| {{y_1}( cdot ) - y( cdot ,t)} right| le d] in a minimum, time. The following maximum principle is proved:Theorem. If [{u^*}(x,t)] is the optimal control and [{t^*}] the optimal time, then there is asolution to the equation[left{ {begin{array}{*{20}{c}}{ - frac{{partial p}}{{partial t}} = sumlimits_{i,j = 1}^n {frac{partial }{{partial {x_i}}}({a_{ji}}(x,t)frac{{partial p}}{{partial {x_j}}}) - alpha p,} }{frac{{partial p}}{{partial {nu _{{A^'}}}}}{|_sum } = 0}end{array}} right.]with the final condition [p(x,{t^*}) = {y^*}(x,{t^*}) - {y_1}(x)], such that[int_Gamma {p(x,t){u^*}} (x,t)dGamma = mathop {max }limits_{u( cdot ) in U} int_Gamma {p(x,t)u(x)dGamma } ] |
