Numerical solution of a parabolic optimal control problem arising in economics and management☆
Introduction
In this article, we construct and study the finite difference approximations of an optimal control problem which is briefly described below.
Consider the pairs satisfying the initial-boundary value problem where is unit outward normal vector to ∂Ω, σi are positive constants and initial value m0(x) is a non-negative function in Ω. We suppose that the function satisfies the boundary condition
Then a solution of problem (1) is non-negative and satisfies mass conservation property:
In what follows the vector-function plays a role of the control function while m(x, t) is a state function in the optimal control problem
The functions and in the definition of the objective functional are supposed to satisfy the following assumptions for all and m ≥ 0:
The form of the objective functional in (5) is motivated by the problems in economics and management (see [8] for 1D problem and [3] for 2D problem). Briefly, the formulation of the model and specifically cost function is driven by problem of modeling the impact of carbon tax and carbon trading on the production behaviors of a very large number of producers. In the model we assume that each produce has an initial emission based on the cap-and-trade mechanism. Producers are pursue the higher level of permitted emission. When the amount of emission is less than permitted level, producer is only paying the corresponding carbon tax. However, when the producer exceeds that level, he has to buy the emission permits for the exceeding part which have higher price than the tax rate.
In our model x1(t) denotes permitted emission, x2(t) is producer’s emission, m(x, t) is the density of producers, and f(x, t, m) is a net revenue of the producer at time t. The assumptions (6) for function f are more general than in the cited articles and allow us to investigate more general applied problems.
The main goal of the paper is the construction and theoretical analysis of the finite-difference approximations of the formulated optimal control problem. We study fully implicit (backward Euler) scheme and two types of fractional steps mesh schemes ([17]) for the state equation. Main feature of these schemes is that their solutions satisfy the discrete analogues of properties (3) and (4) without any constraint for control function and mesh steps in space and in time. Moreover, these solutions are strictly positive for non-negative initial values. This property is the key point in the proof of the existence result for the discrete optimal control problem. Note that in contrast to implicit method and fractional steps methods, the solutions of other well-known approximations of Eq. (1) (explicit, Crank-Nicolson, alternating direction methods) are non-negative only under certain additional constraints connecting max-norm of control function and mesh steps in space and in time. Since the control function is one of the unknowns in the problem, using these difference schemes it may be necessary to a posteriori selection of the corresponding mesh steps in the calculations.
The problems similar to (5) were investigated in [3], [8], [11]. In [8] a theoretical analysis of a differential problem was made and a one-dimensional problem was solved with an explicit approximation of the state equation under corresponding restrictions to the maximum norm of the control function and the mesh step. The analysis and numerous numerical tests for finite difference approximations of one-dimensional problem are presented in [11]. Two-dimensional problem was solved in [3], the state equation was approximated by weighted in time scheme that includes Crank-Nicolson method.
Note that in [3], [8] the authors approximate the system of forward-backward equations obtained from (5) by means of the mean field game theory [12], [13]. In turn, we construct the approximation of (5) itself and deduce the first order optimality conditions by constructing the adjoint state to discrete problem.
The rest of the article is organized as follows.
In the first section we give the formulation of an optimal control problem and prove the existence of its solution. Since the objective functional is not coercive, we prove the existence of a solution under an additional assumption that the control function belongs to a bounded set in the corresponding space.
The second section contains the main results of the article. First, we prove that the matrices of the implicit and fractional steps schemes approximating the state equation are regular irreducible M-matrices, therefore the solutions of these schemes are strictly positive for a non-negative initial data. Then, using this property we prove the existence of the solutions of the corresponding discrete optimization problems without any additional suppositions on the control function (Theorem 2). At the end of the section we propose and study a simple correction procedure for an iterative solution of the implicit scheme to preserve the properties (3) and (4).
In the third section, we derive the first-order optimality conditions for the considered discrete minimization problem in the case of differentiable objective function. The resulting system consists of forward and backward discrete parabolic equations and non-linear equations with multivalued operators. The obtained gradient information is used when solving discrete optimization problem by gradient-type methods.
The last, fourth, section is devoted to the numerical tests. We approximate the state equation by a fractional steps scheme and solve the discrete optimization problem by quasi-Newton method with line search procedure. The numerical tests confirm the effectiveness of the proposed solution methods.
Section snippets
Differential problem
Let us define a weak solution of problem (5). To do this, we introduce several function spaces and list their properties. More information about these spaces can be found in [9], [14], [16].
We use Lebesque spaces Lq(Ω) and Lq(QT) for q ≥ 1, Sobolev space H1(Ω) with the dual (H1(Ω))*. The notation ⟨., .⟩ means pairing between (H1(Ω))* and H1(Ω). The traces of functions from H1(Ω) span Hilbert space H1/2(∂Ω) with dual space the notation ⟨., .⟩γ is used for pairing between and
Approximation of the state equation
Let be a decomposition of into a family Th of nonoverlapping rectangular h1 × h2 finite elements, Vh ⊂ H1(Ω) is the finite element space of the continuous and piecewise bilinear functions (Q1-interpolation, see [4]). By we denote the subspace of Vh × Vh of the vector-functions such thatWe use several quadrature formulas for approximating the integrals of a continuous function g(x). Namely, let b0,0, and
Gradient information and iterative solution methods
Let the function f(x, t, m) have a continuous partial derivative with respect to m for and m > 0. We use the notations and for . It is easy to check that the state functions defined from the equations (21) – (23) are Lipschitz-continuous with respect to therefore, the objective function is also Lipschitz-continuous and has the generalized Clark’s gradient (see [5], [6]).
Below we provide gradient information obtained using
Numerical experiments
To illustrate the application of the methods described above, we performed some numerical experiments. The main purpose of the calculations was twofold: to numerically confirm the theoretical positions on the properties of the solutions of mesh schemes and to evaluate the effectiveness of the applied iterative solution method. We did computational experiments for different meshes, starting with a coarse mesh to a finer one. We observed a good agreement of numerical results both in the optimal
Conclusions
We considered an optimal control problem with a continuous objective function and 2D linear parabolic diffusion-advection state equation, controlled in the coefficient of advection part. The choice of the objective function corresponds to the mathematical models in economics and management. The solutions of the state equation are non-negative and satisfy mass balance equation.
One of the main results of the work was constructing the approximations of the state equation preserving the specified
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Shuhua Zhang was supported by the Natural Science Foundation of China: NSFC 11771322. Alexander Lapin was supported by Russian Foundation of Basic Researches: RFBR 19-01-00431 and by program “1000 talents” of China.