On the Extremum Control Problem with Pointwise Observation for a Parabolic Equation

In this paper we consider a control problem with pointwise observation for a one-dimensional parabolic equation which arises in a mathematical model of climate control in industrial greenhouses. We study a general equation with variable diffusion coefficient, convection coefficient, and depletion potential. For the extremum problem of minimizing an integral weighted quadratic cost functional, we establish the existence and uniqueness of a minimizing function. We also study exact controllability and dense controllability of the problem. Necessary conditions for an extremum are obtained, and qualitative properties of the minimizing function are studied.

We consider the following mixed problem for an equation with a convective term and a depletion potential: (1) (2) (3) where a, b, and h are sufficiently smooth functions in , , (0, T), and . We study a control problem with pointwise observation. Namely, the task is to make the temperature at some point close to a given function on the whole time interval (0, T) by controlling the temperature at the left endpoint of 0 < ( , ) < , a a x t a ϕ ∈ 1 2 (0, ) W T ψ∈ 1 2 W ξ ∈ 2 (0,1) L 0 ( , ) u x t ∈ 0 (0,1) x ∈ 2 ( ) (0, ) z t L T ϕ the interval (the functions and are assumed to be fixed). Let be a set of control functions , and let be a set of target functions z. In what follows, the set is assumed to be nonempty, closed, and convex. The quality of the control is estimated by the cost functional (4) where is the solution of problem (1)-(3) with a given control function and is a weight function such that .
Assuming that the functions z and ρ are fixed, we consider the minimization problem (5) This problem arises in a climate control model for industrial greenhouses (see [1,2]). Detailed explanations concerning the studied mathematical model can be found in [21]. Note that extremum problems for the heat equation have been considered in numerous works (see, e.g., [3][4][5]7]). Problems with final observation have been studied better than others [3][4][5][6]9]. A fairly complete survey of earlier results is given in [6], while more recent results are overviewed in [1,9,10,14,15]. In contrast to previous works on parabolic control problems, which consider problems with a  [5,7,8,11], we consider a pointwise observation. The type of the cost functional is also new. In this paper, we develop and generalize the results of [16][17][18][19][20][21]. Specifically, we study a more general equation in which the diffusion a and convection b coefficients and the depletion potential h are all variable and establish qualitative properties of the corresponding minimizing function. In addition to the study of a more general equation (with  variable coefficients and and an inhomogeneous initial condition), some new results are proved, namely, we establish qualitative properties of the minimizing function and derive necessary optimality conditions. The proof relies on the results and methods presented in [12,13]. , which satisfies the inequality where C 1 is a constant independent of , , and . Corollary 1. The mapping from the space to is continuous.
To derive the estimate presented below, we need the following positivity principle. To prove Theorem 2, we change the unknown function, so problem (1) with a Robin boundary condition. Next, the required result is obtained by applying a modified barrier function method.
With the use of Theorem 2, we obtain the following estimate.  , it is true that , ρ, Φ]. An important issue is the exact controllability of the extremum problem.
The function is then called an exact control.
The following theorem states that the set Z of functions admitting exact controllability is a sufficiently "small" subset of . Theorem 7. The set Z of all functions admitting exact controllability, i.e., such that = 0 for some , is a first-category set in . Now we examine the dense controllability of the problem. Another important issue is one of obtaining necessary minimum conditions for . A necessary condition can be formulated in terms of the conjugate problem for (1)-(3), (5). By the conjugate problem, we mean the following mixed problem for an inverse parabolic equation: (14) (15) (16) where is a solution of (1)-(3).
Definition 5. A solution of problem (15)-(17) is a function satisfying the condition = 0 and the integral identity (17) for all functions such that and . Theorem 9. Problem (15)-(17) has a unique solution , which satisfies the inequality   (19) where p is the solution of problem (15)- (17) with .
Note that the trace of the derivative exists by the theorem on regularity of solutions to parabolic boundary value problems (see [23,Chap. 3,Sect. 12]).
A necessary condition for a function to be a minimizer can also be obtained without using the conjugate problem. For example, if is a minimizing function, then, for any control function , we have T u x t z t u x t u x t t dt