CLASSICAL SOLUTIONS OF LINEAR REGULATOR FOR DEGENERATE DIFFUSIONS

We study a stochastic control problem for linear degenerate systems. We establish the existence of a classical solution of the degenerate Bellman equation by the technique of viscosity solutions, and the optimal policy is shown to exist from the optimality conditions.


Introduction
The usual framework of control is the one given in probably the most studied control problem, stochastic regulator control problem, which deals with minimizing a performance index of a system governed by a set of differential equations. The stochastic linear regulator problem has been studied by many authors including Bensoussan [4], Fleming and Soner [9] for nondegenerate diffusions. Da Prato [8] gives the solution to the stochastic linear regulator for the degenerate systems related to Riccati equations (i.e., any ordinary differential equation) for the quadratic case with infinite horizon. But he has not established the existence of a classical solution of the Hamilton-Jacobi-Bellman (HJB) equation (i.e., a partial differential equation in the optimal control theory) to the linear regulator control problem. Here we have studied an extended stochastic control problem of the linear regulator for the degenerate diffusions by considering the general case with infinite horizon.
We are concerned with the stochastic control problem to minimize the discounted expected cost: 2 Classical solutions of linear regulator for degenerate diffusions over c ∈ Ꮽ subject to the degenerate stochastic differential equation dx t = Ax t + c t dt + σx t dw t , x 0 = x ∈ R, t ≥ 0, (1.2) for α > 0, m ≥ 0, nonzero constants A, σ = 0, and a continuous function f on R such that (1.4) for some constants K, k 0 , k 1 > 0. Here, (w t ) is a one-dimensinal standard Brownian motion on a complete probability space (Ω,Ᏺ,P) endowed with the natural filtration Ᏺ t generated by σ(w s ,s ≤ t), and Ꮽ denotes the class of all Ᏺ t -progressively measurable pro- We also refer to Bensoussan [4], Fleming and Soner [9] for nondegenerate diffusions, and also Da Prato [8] for the degenerate stochastic system from the view of Riccati equations in case of f (x) = Kx 2 and m = 2.
The main purpose of the linear regulator problem (1.1) and (1.2) is to give a synthesis of optimal control for degenerate stochastic systems by a classical solution u of the associated HJB equation where α > 0, u x , u xx are partial derivatives of u(x,t) with respect to x. Generaly speaking, the difficulty stems from the degeneracy in the second-order term of (1.5).
Our objective is to find the viscosity solution for u of (1.5) following Bardi and Capuzzo-Dolcetta [2], Crandall et al. [6], Fleming and Soner [9] through the limit of the solution v = v L , L > 0, to the HJB equation as L → ∞, where the value function v L can be defined as a function whose value is the minimum value of the objective function of the control problem for the system, that is, where Ꮽ L = {for all c = (c t ) ∈ Ꮽ such that |c t | ≤ L for all t ≥ 0}. Also our study deals with the smoothness of the viscosity solution u of (1.5) using a convexity argument of v L (x) and u(x).
To this end, we assume that f is uniformly continuous with m-weight, that is, there exists C ρ > 0, for any ρ > 0, such that (1.8) We notice that (1.8) is fulfilled for f (x) = |x| μ , 0 ≤ μ ≤ m, and plays an important role as treated in Koike and Morimoto [12], Menaldi and Robin [14].
In Section 2, we show that the value function u(x) := lim L→∞ v L (x) is a viscosity solution of (1.5). Section 3 is devoted to the study of smoothness of u. Finally, in Section 4, we present an optimal control of the control problem (1.1) and (1.2).

Viscosity solutions
We studied here the properties of the value function v L (x) using the method of dynamic programming, initiated by Bellman [3] and and showed that v L (x) converges to a viscosity solution u(x) of the Bellman equation (1.5).
The notion of viscosity solutions to HJB equation was introduced by Crandall and Lions [7] in the early 80's and requires only the solution to be continuous and by Lions [13] for second-order equations. Crandall et al. [6] showed a modern presentation of this notion of solution in the User's Guide to viscosity solutions and Fleming and Soner [9] also described the connections with control problems. Bardi and Capuzzo-Dolcetta [2] showed an introduction to the theory limited to first-order equations. They also characterized the value function as a viscosity solution of HJB equation.
Let ω : R → R be a scalar function, defined on an open set R ⊆ R 2 . In the following, we consider the second-order, partial differential equation The set of superdifferentials of ω at a point x is defined as Similarly, the set of subdifferentials of ω at a point x is defined as where (·) stands for the scalar product of two vectors in R 2 . We recall by Crandall et al. [6] the definition of viscosity solutions of (2.1) in terms of sub-and superdifferentials.
Similarly, a function ω ∈ C(R) is called a viscosity supersolution of (2.1) if is a viscosity solution of (2.1) if it is both viscosity sub-and supersolutions of (2.1). We now turn to the second definition of a viscosity solution which is equivalent to the previous one.

Classical solutions of linear regulator for degenerate diffusions
Equivalent definition. A function ω ∈ C(R) is called a viscosity subsolution of (2.1) if, for every ϕ ∈ C 2 (R), ω − ϕ has a local maximum at x ∈ R, then Similarly, a function ω ∈ C(R) is called a viscosity supersolution of (2.1) if, for every In order to ensure the integrability of J(c), we assume that for some constant κ μ > 0, depending on μ.
Proof. Let (z t ) be the unique solution of This yields that such c belongs to C L , and thus Let (y t ) be the solution y t of (1.2) with y 0 = y. It is clear that x t − y t fulfills (2.17) with initial condition x − y. Hence, by (1.8), (2.21), and Lemma 2.2, where C ρ = C ρ κ m and ρ = ρ/(1/α + 2θ). Therefore, we deduce (2.16), completing the proof.
Proof. We denote v(x) the right-hand side of (2.23). By the formal Markov property, with c equal to c shifted by τ. Thus Md. Azizul Baten 7 It is known in Fleming and Soner [9], Nisio [15] that this formal argument can be verified, and we deduce v L (x) ≥ v(x).
To prove the reverse inequality, we take 0 < δ < 1 with C ρ δ m < ρ for any ρ > 0. Then, by (2.22), we have for |x − y| < δ, (2.27) By (2.21), for any i, we take x (i) ∈ S i and c (i) ∈ Ꮿ L such that On the other hand, by the definition of v(x), we can find c ∈ Ꮽ L such that Then we have   (Apostol [1]), we can observe the locally uniform convergence and the viscosity property of u (Crandall et al. [6]). The proof is complete.

Classical solutions
We studied the smoothness of the viscosity solution u of (1.5). In what follows, we say that u is a classical solution of (1.5) if it is twice differentiable and satisfies the equation pointwise. The value function, in general, is not smooth even for smooth systems. In order to prove the smoothness of the viscosity solution u of (1.5), we used a convexity argument of v L (x), u(x) and the technique of viscosity solutions is used to construct solutions.

Proof
Step 1. By the convexity of u, we recall a classical result of Fleming and Soner [9] to see that lebesgue measure of R \ Ᏸ ∪ {0} = 0, where By the definition of twice differentiability, we have and hence −αu + 1 2 Let d + u(x) and d − u(x) denote the right-and the left-hand drrivatives of u(x), respectively. Define r ± (x) by (3.5) Since d + u = d − u = u on Ᏸ, we have r + = r − = u a.e. By definition, d + u(x) is right continuous, and so is r + (x). Hence it is easy to see that (3.6) Thus we get (3.7) Step 2. We claim that u(x) is differentiable at x ∈ R \ Ᏸ ∪ {0}. It is well known in Bardi and Capuzzo-Dolcetta [2], Clarke [5] that then we can find a sequence y n → x such that lim n→∞ R(u; y n ) < 0. By (3.7), we may consider that y n ≤ y n+1 < x for every n, taking a subsequence if necessary. Hence lim n→∞ u y n − u(x) − d + u(x) y n − x y n − x ≤ 0, (3.11) this leads to d + u(x) ≤ d − u(x), which is a contradiction. Thus we have (d + u(x),r + (x)) ∈ J 2,− u(x) and similarly, (d − u(x),r − (x)) ∈ J 2,− u(x). By the convexity of J 2,− u(x), we get ( p, r) ∈ J 2,− u(x). Now we note that (3.12) and hence by (3.5), On the other hand, by the definition of viscosity solution, which is a contradiction. Therefore, we deduce that ∂u(x) is a singleton, and so u is differentiable at x Clarke [5].
Step 3. We claim that u is continuous on (R \ {0}). Let x n → x and p n = u (x n ) → p.
Then by convexity, we have Step 4. We set w = u . Since Then In addition, if f = 0, then Proof. We first observe that v L is a viscosity solution of the boundary value problem: Substituting y = 2x and y = 0, we get u(2x   The proof is complete.