Min-max game theory and nonstandard differential Riccati equations for abstract hyperbolic-like equations |
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Authors: | R. Triggiani |
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Affiliation: | Department of Mathematics, University of Virginia, Charlottesville, VA 22903, United States Department of Mathematics and Statistics, KFUPM, Dhahran, 31261, SA, United States |
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Abstract: | We consider the abstract dynamical framework of Lasiecka and Triggiani (2000) [1, Chapter 9], which models a large variety of mixed PDE problems (see specific classes in the Introduction) with boundary or point control, all defined on a smooth, bounded domain Ω⊂Rn, n arbitrary. This means that the input → solution map is bounded on natural function spaces. We then study min-max game theory problem over a finite time horizon. The solution is expressed in terms of a (positive, self-adjoint) time-dependent Riccati operator, solution of a non-standard differential Riccati equation, which expresses the optimal qualities in pointwise feedback form. In concrete PDE problems, both control and deterministic disturbance may be applied on the boundary, or as a Dirac measure at a point. The observation operator has some smoothing properties. |
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Keywords: | Min-Max game theory Non-standard Riccati equations Hyperbolic dynamics |
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