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Time-optimal control of parabolic time lag system. (English) Zbl 0831.49008

The linear parabolic equation with Neumann boundary condition is considered. The control is included in the constant term. It is necessary to find the control from a convex set which transfer the system in a given neighbourhood of some final state in the course of minimum time.
The existence of this problem and the necessary conditions of optimality were proved earlier by knowles under non-triviality of the problem. In this paper, the transformation of those results is given by means of the adjoint system. The bang-bang principle is proved.
By the way the similar transformation is known [see, for exemple, J.-L. Lions: “Optimal control of systems governed by partial differential equations” (1991; Zbl 0203.090)]. It seems doubtful the statement of the authors that this results can be extended directly to nonlinear systems. The linearity of the equation is necessary in this method. The nonlinear problems can be investigated by means of quite other methods.
It is unknown if the non-triviality of the optimal control problem has to do with controllability. It is interesting to find the influence of the radius of the ball \(\varepsilon\) to the conditions of optimality. For example if the limit for \(\varepsilon\to 0\) exists then the exact controllability can be proved. Then it is possible to solve the time- optimal control problem of the transfer of the system in the given final state.

MSC:

49J25 Optimal control problems with equations with ret. arguments (exist.) (MSC2000)
49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations

Citations:

Zbl 0203.090
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