Z. Tsintsadze

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

OPTIMIZATION OF LINEAR CONTROLLED SYSTEMS WITH RESTRICTIONS OF “NARROW PLACE” TYPE

Consider an industrial process (see [1]), which is described with the linear system of differential equations

ẋ_{1} = u_{1}, ẋ_{2} = u_{2}, ẋ_{3} = u_{3} − b_{1}u_{1} − b_{2}u_{2} − b_{4}u_{4}, ẋ_{4} = u_{4} (1)

under the initial conditions

x_{1}(0) = c_{1}, x_{2}(0) = c_{2} > 0, x_{3}(0) = c_{3}, x_{4}(0) = c_{4} > 0,  (2)

where x_{1}(t) is the reserve of motor-car in moment t, x_{2}(t) is the industrial power of motor-car branch in t, x_{3}(t) is the reserve of steel in moment t, x_{4}(t) is the industrial power of steelfoundry branch in moment t, b_{i}, c_{i} are given constants and u_{i} = u_{i}(t), i = \overline{1, 4}, are the control functions.

We consider the optimal control problem for this process in the form:

x_{1}(T) → Sup  (3)

under the restrictions

u_{1} − x_{2} ≤ 0, u_{3} − x_{4} ≤ 0, u_{j−2} ≤ 0, j = 3, 4, 5, 6, x_{3} > 0 (4)

and conjunctions (1), (2). Here T is fixed, u_{1} is a speed of motor-car production, u_{2} is a speed of industrial power rising of motor-car branch, u_{3} is a speed of steelproduction, u_{4} is a speed of industrial power rising of steelfoundry branch. In different from general problems, which are considered in common theory from [2], the given problem with restrictions of “narrow place” type is characteristed.

Let x_{i}(t), i = \overline{1, 4}, be absolutely continuous, u_{i}(t), i = \overline{1, 4}, be integrable functions on [0, T], and the conditions (1), (4) be fulfilled almost everywhere on [0, T], then using the Theorem 1 from [3], we obtain the necessary conditions of optimality for problem (1)–(4).

Detailed analysis of the behaviour of system (1)–(4) when satisfying the necessary conditions of optimality enables one to optimize this system for different corelations between parameters b_{1}, b_{2}, b_{4} and T. For example, if T < b_{4}, then the optimal is such a behaviour of the system at which the industrial power of both branches remain constant and are used maximally.

References

1. R. Bellman, Dynamic programming. M, 1960.

2. R. V. Gamkrelidze and G. L. Kharatishvili, Extremal problems in linear topological spaces. J. Math. Systems Theory 1(1967), No. 3, 229–256.

3. Z. Tsintsadze, The Lagrange principle of taking restrictions off and its appl. Mem. Differential Equations Math. Phys. 11(1997), 105–128.