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Some remarks on the existence of optimal controls for quasilinear systems

Abstract

Let a quasilinear control system having the state space\(\bar X \subseteq R^n \) be governed by the vector differential equation

$$\dot x = G(u(t))x,$$

wherex(0) =x 0 andU is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR m.LetG:UR be a bounded measurable nonlinear function, such thatG(U) is compact and convex.G −1 can be convex onG(U) or concave. The main results of the paper establish the existence of a controluU which minimizes the cost functional

$$I(u) = \int_0^T {L(u(t))x(t)dt,} $$

whereL(·) is convex. A practical example of application for chemical reactions is worked out in detail.

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References

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    Socha, L., andSkrzypek, J.,Application of Open-Loop Control to the Determination of the Optimal Temperature Profile in a Chemical Reactor, Proceedings of 8th IFIP Conference on Optimization Techniques, Würzburg, Germany, 1977.

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    Boyarsky, A.,On the Existence of Optimal Controls for Nonlinear Systems, Journal of Optimization Theory and Applications, Vol. 20, No. 2, 1976.

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    Friedland, S.,Global Principle for Free-Endpoint Problems in Optimal Control and Differential Games, Journal of Optimization Theory and Applications, Vol. 24, No. 2, 1978.

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Communicated by G. Leitmann

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Socha, L. Some remarks on the existence of optimal controls for quasilinear systems. J Optim Theory Appl 33, 393–399 (1981). https://doi.org/10.1007/BF00935251

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Key words

  • Optimal control
  • nonlinear systems
  • existence theorems