| Abstract: | In this paper we consider the systems governed, by parabolioc equations
\\frac{{\partial y}}{{\partial t}} = \sum\limits_{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 a
solution to the equation
\\left\{ {\begin{array}{*{20}{c}}
{ - \frac{{\partial p}}{{\partial t}} = \sum\limits_{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)d\Gamma = \mathop {\max }\limits_{u( \cdot ) \in U} \int_\Gamma {p(x,t)u(x)d\Gamma } \] |
