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We study the ergodic control problem related to stochastic production planning in a single product manufacturing system with production constraints. The existence of a solution to the corresponding Hamilton-Jacobi-Bellman equation and its properties are shown. Furthermore, the optimal control for the ergodic control problem and an example are given.

This paper deals with the following first-order nonlinear differential equation: -i~(x,i)+Aw(x,i)+h(x), x E R1, i=1,2 )...) d. (1) Here, X is a constant, F(x) = kx if x < 0, = 0 if x 2 0 for some positive constant k > 0, h is a convex function, and A denotes the infinitesimal generator of an irreducible Markov chain (z(t), P) with state space 2 = {1,2,. . . , d}, A+, i) = c q&(x,j) - v(x, i)], j#i (2) where qij is the jump rate of z(t) from state i to state j. The unknown is the pair (II, A), where w(.,i) E C’(R1) for every i E 2. Equation (1) arises in the ergodic control problem of stochastic production planning in a single product manufacturing system and is called the Hamilton-Jacobi-Bellman (HJB) equation or dynamic programming equation. The inventory level s(t) of stochastic production planning modelled by Sethi and Zhang [l, Section 3.5, p. 50) is governed by the differential equation dx(t) - = p(t) - da dt x(0) = 2, t(0) = i, P-a.s.,for production rate 0 I p(t) 5 k, in which z(t) and k are interpreted as the demand rate and capacity level, respectively. For the ergodic control problem, the cost J@(e) : rc,i) associated with p(e) is given by J(p(.) : z, i) = lim sup LE T-co T l+(t)) dt 1 z(O) = 2, z(0) = i 1 , where h(z) represents the convex inventory cost. The purpose of this paper is to show the existence of a solution to the HJB equation (1) and to present the optimal control minimizing the cost J@(e) : z, i) subject to (3). In the control problem of stochastic production planning with discounted rate a > 0, Bensoussan et al. [2], Fleming et al. [3], and Sethi et al. [4] have investigated the HJB equation For the ergodic control problem, we study the limit of (5) as a tends to 0, and investigate a solution to the degenerate HJB equation (1). This approach develops the technique of Bensoussan and Frehse [5] concerning nondegenerate second-order partial differential equations to our degenerate case. Section 2 is devoted to the existence problem of the HJB equation (1) under the convexity assumption and others on h, and properties of the solution are shown in Section 3. In Section 4, an optimal control for the ergodic control problem and the value are given. In Section 5, we present an example of the solution to the HJB equation (l), and the optimal control and the value are given.

In this paper, we have examined the ergodic control problem for the stochastic production planning with production constraints. We show that there exists a solution (v, A) to the corre- sponding degenerate HJB equation under some conditions, and properties of the solution (v, A) are investigated. Moreover, the optimal control p* for the ergodic control problem is given in term of the partial derivative $$ of the solution (w,X) to the HJB equation, and the value is given by constant X of the solution (u, A). Finally, more work needs to show the uniqueness of the solution to the HJB equation.