Finite Time--Horizon Risk Sensitive Control and the Robust Limit under a Quadratic Growth A.pdf

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Finite Time--Horizon Risk Sensitive Control and the Robust Limit under a Quadratic Growth A.pdf

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Finite Time--Horizon Risk Sensitive Control and the Robust Limit under a Quadratic Growth A

Finite Time{Horizon Risk Sensitive Control and the RobustLimit under a Quadratic Growth AssumptionFrancesca Da Lio William M. McEneaneyyOctober 21, 1998AbstractThe nite time{horizon risk sensitive limit problem for continuous, nonlinear systems is consi-dered. Previous results are extended to cover more typical examples. In particular, the cost maygrow quadratically, and the diusion coecient may depend on the state. It is shown that the risksensitive value function is the solution of the corresponding dynamic programming equation. It isalso shown that this value converges to the value of the Robust control problem as the cost becomesinntely risk averse with corresponding scaling of the diusion coecient.Key Words: risk-sensitive control, robust, H1 , viscosity solutions, nonlinear HJB equations, nonlin-ear Isaacs equationsAMS subject classications: 35B37, 49L25, 90D25, 93B36, 93C10, 93E05, 93E201 IntroductionThe nonlinear, nite time{horizon risk sensitive limit problem is considered. It is, by now, well known thatthe value functions of risk{sensitive stochastic control problems tend to converge to the value functionsof the corresponding Robust/H1 control problems as one approaches innite risk aversion. This wasaddressed rst in the LEQG (linear{exponential{quadratic{gaussian) case where, in fact, one does notneed to take the risk averse limit [23], [42], [5],[9]. In the nonlinear, case, results were rst developedfor nite time{horizon control problems [43],[24], [14], [32], [33]. Further studies considered nonlinear,innite time{horizon problems (in which case one gets the H1 limit) [15], [33], [16], [13], [36], [18], [38],[28], [34], [4] and nonlinear escape problems [27], [7]. Other studies have involved discrete systems [39],[10], [12], and the partial observations case [6], [25].The results for both the nite time{horizon problem and the innite time{horizon problem have mainlybeen obtained under assumptions which preclude quadratic cost criteria. In o