Dynamic programming and maximum principle for discrete Goursat systems

Existence and Uniqueness of Solutions for two Classes of Functional Equations Arising in Dynamic Programming

“…In 1984, Bhakta and Mitra [7] investigated the existence and uniqueness of solutions for the functional equation (1.1) with opt = sup. In 1988 and 1991, Bhakta and Choudhury [6] and Belbas [1] obtained the existence and uniqueness of solutions for the functional equation (1.1) with opt = inf and p = 0. In 2001, Liu [13] studied the existence, uniqueness and iterative approximation of solutions for the functional equation (1.1) in BB(S), see Section 2.…”

“…The functional equation (1.1) and its special cases have been studied by many researchers, see, for example, [1][2][3][4][5][6][7] and [9][10][11][12][13][14][15][16][17][18][19][20] and the references therein. Bellman [2,3] and Bellman and Roosta [5] established the existence and iterative approximation of solutions for some functional equations that are special cases of (1.1).…”

On solvability of functional equations and system of functional equations arising in dynamic programming

“…This paper deals with solvability of functional equations and system of functional equations arising in dynamic programming of multistage decision processes as follows: studied by Bellman [3,4], Bellman and Lee [5], Bellman and Roosta [6], Bhakta and Choudhury [7], Bhakta and Mitra [8], Liu [12], Liu and Ume [13], and others [1,2], respectively; the systems of functional equations f (x) = sup y∈D {u(x, y) + G(x, y, g(a(x, y)))}, ∀x ∈ S, g(x) = sup y∈D {u(x, y) + F (x, y, f (a(x, y)))}, ∀x ∈ S, (1.11) f (x) = inf y∈D {v(x, y) + G(x, y, g(a(x, y)))}, ∀x ∈ S, g(x) = sup y∈D {u(x, y) + F (x, y, f (b(x, y)))}, ∀x ∈ S, (1.12) investigated by Chang [10], Chang and Ma [11], and Liu [12], respectively. By using fixed point theorems, we establish the existence and uniqueness of solutions and iterative approximation for the functional equations (1.1) and (1.2).…”

Dynamic programming for 2D discrete nonlinear systems

“…Theorem 1: Assume that u(t, s), (t, s) ∈ T × S is an admissible control function and x(t, s) is a corresponding solution of (1) and (2). Then {u(t, s), x(t, s), (t, s) ∈ T × S } is optimal if, and only if, for each t = 1, 2, ...,t 1 , the following 978-1-4244-2798-7/09/$25.00 ©2009 IEEE conditions hold…”

“…The subject of this paper is the use of a dynamic programming approach to solve a 2D discrete nonlinear systems optimization problem previously considered in [1] (see, also, [2]). The first results develop necessary and sufficient optimality conditions and can be regarded as the extension of Bellman equations introduced for the standard, or 1D case, in [3] to 2D system.…”