Backward Stochastic Riccati Equations and Infinite Horizon L-Q Optimal Control with Infinite Dimensional State Space and Random Coefficients |
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Authors: | Giuseppina Guatteri Gianmario Tessitore |
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Institution: | (1) Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy;(2) Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via R. Cozzi 53–Edificio U5, 20125 Milan, Italy |
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Abstract: | We study the Riccati equation arising in a class of quadratic optimal control problems with infinite dimensional stochastic
differential state equation and infinite horizon cost functional. We allow the coefficients, both in the state equation and
in the cost, to be random.
In such a context backward stochastic Riccati equations are backward stochastic differential equations in the whole positive
real axis that involve quadratic non-linearities and take values in a non-Hilbertian space. We prove existence of a minimal
non-negative solution and, under additional assumptions, its uniqueness. We show that such a solution allows to perform the
synthesis of the optimal control and investigate its attractivity properties. Finally the case where the coefficients are
stationary is addressed and an example concerning a controlled wave equation in random media is proposed. |
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Keywords: | Infinite horizon Backward stochastic Riccati equation Linear quadratic optimal control Stochastic coefficient |
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