JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 119, 234-248 (1986) On Bellman Equations in Duality A. BENSOUSSAN University Paris-Dauphine and INRIA, 78153 Le Chesnay Cedex, France Submitted by E. Stanley Lee Received March 7, 1986 DEDICATED TO THE MEMORY OF RICHARD BELLMAN INTRODUCTION A prototype of a dual Bellman equation is the following: find a function $(<) solution of: W&W, VI- 4 . sW, v)l = 0, a.e. 5 (*I ” which compares to the usual Bellman equation: find x(x) satisfying: Inf[4(x, u) + & .g(x, v)] = 0, a.e. x. (**I One derives (**) from (*) by the Legendre transformation: 5+Dx(x)=O w(r) = x. Dual Bellman equations come out naturally in the study of boundary layers related to singular perturbations in optimal control (see [ 11). In fact, this is a generalization of a correspondence known in the case of linear quadratic problems, involving singular perturbations (see [3], for instance). The corresponding Riccati equations also play a role in the duality between filtering and control (see [2]). The classical Bellman equation (w) has not a unique solution, even in the class of regular solutions, leaving aside that the solution is defined up to a constant. This is already the case for the linear quadratic problems in which (**) reduces to a Riccati equation. This caseis presented in Section 1, although classical; it corresponds basically to the positive real lemma. A complete theory can be found in the above book. 234 0022-247X/86 $3.00 Copynght 0 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.