Control Problem for Dynamical Systems with Partial Derivatives
- PDF / 239,456 Bytes
- 13 Pages / 594 x 792 pts Page_size
- 74 Downloads / 184 Views
Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
CONTROL PROBLEM FOR DYNAMICAL SYSTEMS WITH PARTIAL DERIVATIVES S. P. Zubova ∗ Voronezh State University 1, Universitetskaya pl., Voronezh 394006, Russia [email protected]
E. V. Raetskaya Voronezh State University of Forestry and Technologies 8, Timiryazeva St., Voronezh 394613, Russia [email protected]
Le Hai Trung University of Danang 41, Leduan St., Danang 550000, Vietnam [email protected]
UDC 517.95
Using the cascade spitting method, for a linear dynamical system with partial derivatives we construct a control function under the action of which the system is transferred from an arbitrary initial state to an arbitrary final state. For such systems we obtain the full controllability criterion. The control and state functions are found in an analytical form. Bibliography: 10 titles. Illustrations: 2 figures.
1
Introduction
We consider the equation ∂y ∂y (1.1) =B + Du(t, x), t ∈ [0, T ], x ∈ [0, xk ], ∂t ∂x where y = y(t, x) ∈ Rn , u = u(t, x) ∈ Rm , and B, D are matrices of appropriate sizes. The system (1.1) is fully controllable if there exists a control u(t, x) under the action of which the system (1.1) goes from an arbitrary sufficiently smooth initial state y(0, x) = α(x) ∈ Rn
(1.2)
to an arbitrary sufficiently smooth final state y(T, x) = β(x) ∈ Rn ∗
(1.3)
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 113-123. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0941
941
during the time interval [0, T ], where T > 0 [1]. A certain smoothness condition on the vector-valued functions α(x) and β(x) can be necessary for the full controllability of the system. For example, if ∂y1 (t, x) = u(t, x), ∂t
∂y2 (t, x) ∂y1 (t) = , ∂t ∂x
α(x) = (α1 (x), α2 (x)),
then the second equation takes the form ∂α1 (x) ∂y2 (t, x) = ∂t ∂x t=0 ∂y2 (t, x) ∂β1 (x) = ∂t ∂x t=T
β(x) = (β1 (x), β2 (x)),
at t = 0, at t = T .
Thus, the differentiablity of α(x) and β(x) is necessary. We are interested in finding conditions on B and D guaranteeing the full controllability of the system (1.1), constructing u(t, x) and y(t, x) in an analytical form, and determining sufficient smoothness conditions on α(x) and β(x) that ensure the full controllability of the system. For this purpose the cascade splitting method (or the cascade method) developed in [2]–[9] is first used for the systems under consideration. The method consists in constructing u(t, x) and y(t, x) in an analytical form. It is not required to verify preliminarily the full controllability of the system. Whether the system is fully controllable or not will be clear in the process, as well as the differentiability orders of α(x) and β(x) sufficient to apply this method. Nevertheless, owing to the full controllability criterion for the system (1.1) and sufficient smoothness conditions on α(x) and β(x), we can formulate the method in terms of the coefficients B and D. In the process of cascade decomposition, the nonuniqueness of the
Data Loading...
