Control Problem

Generally speaking, temperature control in fixed beds is difficult because heat loads vary through the bed. Also, in exothermic reactors, the temperature in the catalyst can become locally excessive. Such hot spots can cause the onset of undesired reactions or catalyst degradation. In tubular devices such as shown in Fig. 2.6a and b, the smaller the diameter of tube, the better is the temperature control. Temperature-control problems also can be overcome by using a mixture of catalyst and inert solid to effectively dilute the catalyst. Varying this mixture allows the rate of reaction in different parts of the bed to be controlled more easily. 

Thus, the function / solves the optimal control problem (2.19). Theorem 2.2 is proved. 

Different optimal control problems can be found in the monographs and papers (Khludnev, Sokolowski, 1997 Banichuk, 1980 Barbu, 1984 Cea, 1971 Lions, 1968a, 1968b Litvinov, 1987 Mignot, 1976 Fuel, 1987 Bock, Lovisek, 1987, Haslinger et ah, 1986). 

As we know the vertical displacements of the plate defined from (2.7), (2.8) can be found as a limit of solutions to the problem (2.9)-(2.11). Two questions arise in this case. The first one is the following. Is it possible to solve an optimal control problem like (2.19) when w = w/ is defined from (2.9)-(2.11) The second question concerns relationships between solutions of (2.19) and those of the regularized optimal control problem. Our goal in this subsection is to answer these questions. 

First of all let us formulate the regularized optimal control problem. If the set F is introduced in similar way and w/ = w is found from the equation... 

Theorem 2.7. Under the above conditions, there exists a solution of the optimal control problem (2.49). 

In the sequel we shall study an optimal control problem. Let C (fl) be a convex, bounded and closed set. Assume that ( < 0 on T for each G. In particular, this condition provides nonemptiness for Kf. Denote the solution of (2.131) by % = introduce the cost functional... 

In the next two subsections the parameter c is supposed to be fixed. The convergence of solutions of the optimal control problem (2.134) as —> 0 will be analysed in Section 2.5.4. For this reason the -dependence of the cost functional is indicated. 

Let the set be the same as in Section 2.5.2. Consider the optimal control problem... 

Suppose that 5 is fixed for the time being. We shall prove that a solution of the optimal control problem (2.189), (2.188) exists. We choose a minimizing sequence Um U. It is bounded in and so we can assume... 

An existence theorem to the equilibrium problem of the plate is proved. A complete system of equations and inequalities fulfilled at the crack faces is found. The solvability of the optimal control problem with a cost functional characterizing an opening of the crack is established. The solution is shown to belong to the space C °° near crack points provided the crack opening is equal to zero. The results of this section are published in (Khludnev, 1996c). 

Further, in Section 3.1.4, an optimal control problem is analysed. The external forces u serve as a control. The solution existence of the optimal control problem with a cost functional describing the crack opening is proved. Finally, in Section 3.1.5, we prove C°°-regularity of the solution near crack points having a zero opening. 

The goal of this subsection is to prove an existence theorem for the optimal control problem. 

The result given below provides the solvability of the optimal control problem formulated. 

In this subsection we analyse an optimal control problem. The exterior forces f,g) are chosen to minimize the cost functional... 

This functional characterizes an opening of the crack. As before, x,0 is the solution of (3.48) corresponding to f,g)- At the first step we prove the existence of the optimal control problem. The next step is to prove the C°°-regularity of the solution provided that the crack opening is zero. We fixed the parameter c in this subsection the passage to the limit, as c —> 0, is analysed in Section 3.2.4. 

Consider an approximate description of the nonpenetration condition between the crack faces which can be obtained by putting c = 0 in (3.45). Similar to the case c > 0, we can analyse the equilibrium problem of the plates and prove the solution existence of the optimal control problem of the plates with the same cost functional. We aim at the convergence proof of solutions of the optimal control problem as —> 0. In this subsection we assume that T, is a segment of a straight line parallel to the axis x. 

The optimal control problem to be analysed is formulated as follows to find an element 0 6 such that... 

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