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Mat. Zametki, 2006, Volume 80, Issue 6, Pages 885–901 (Mi mz3364)  

Optimal control in nonlinear infinite-dimensional systems with nondifferentiability of two types

S. Ya. Serovaĭskiĭ

Al-Farabi Kazakh National University

Abstract: We consider the optimal control problem for systems described by nonlinear equations of elliptic type. If the nonlinear term in the equation is smooth and the nonlinearity increases at a comparatively low rate of growth, then necessary conditions for optimality can be obtained by well-known methods. For small values of the nonlinearity exponent in the smooth case, we propose to approximate the state operator by a certain differentiable operator. We show that the solution of the approximate problem obtained by standard methods ensures that the optimality criterion for the initial problem is close to its minimal value. For sufficiently large values of the nonlinearity exponent, the dependence of the state function on the control is nondifferentiable even under smoothness conditions for the operator. But this dependence becomes differentiable in a certain extended sense, which is sufficient for obtaining necessary conditions for optimality. Finally, if there is no smoothness and no restrictions are imposed on the nonlinearity exponent of the equation, then a smooth approximation of the state operator is possible. Next, we obtain necessary conditions for optimality of the approximate problem using the notion of extended differentiability of the solution of the equation approximated with respect to the control, and then we show that the optimal control of the approximated extremum problem minimizes the original functional with arbitrary accuracy.

DOI: https://doi.org/10.4213/mzm3364

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English version:
Mathematical Notes, 2006, 80:6, 833–847

Bibliographic databases:

UDC: 517.977
Received: 22.07.2004
Revised: 13.02.2006

Citation: S. Ya. Serovaǐskiǐ, “Optimal control in nonlinear infinite-dimensional systems with nondifferentiability of two types”, Mat. Zametki, 80:6 (2006), 885–901; Math. Notes, 80:6 (2006), 833–847

Citation in format AMSBIB
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