On Krasovskii Criterion of Optimal Stabilization

Oleksiy Ignatyev Kent State University, USA

Consider a system of differential equations of perturbed motion

ẋ = X(t, x; u)  (1)

where x = (x_{1}, ..., x_{n}), X = (X_{1}, ..., X_{n}), u = (u_{1}, ..., u_{r}) -is a control function, and X(t, x; u)-is an almost periodic function in t.

Suppose that function X(t, x; u) is defined, continuous and satisfying Lipschitz condition in x in the domain

t ∈ R, ∥x∥ < H (H = const).  (2)

Suppose that criterion of motion x(t) is given as

I = ∫_{t_{0}}^{∞} ω(t, x_{1}[t], ..., x_{n}[t]; u_{1}[t], ..., u_{r}[t]) dt,  (3)

where ω(t, x, u) is nonnegative function.

Theorem. If there exist almost periodic in t, positive definite, continuously differentiable function V^{0}(t, x) and functions u_{j}^{0}(t, x), which are satisfying in the domain (2) the following conditions:

1) function w(t, x) = ω(t, x; u^{0}(t, x)) is nonnegative, and w(t, x) may equal zero only in the points of set which does not include any samitrajectory of the system (1) x(x_{0}, t_{0}, t), (t_{0} < t < +∞) entirely (except the trivial solution);

2) an equality

B[V^{0}; t, x; u^{0}(t, x)] = 0  (4)

holds, where

B[V; t, x; u] = \frac{∂V}{∂t} + ∑_{i=1}^{n} \frac{∂V}{∂x_{i}} X_{i}(t, x; u) + ω(t, x; u) = \frac{dV}{dt} + ω(t, x; u); (5)

3) an equality

B[V^{0}; t, x; u] ≥ 0,  (6)

holds for each control functions u_{j}, then functions u_{j}^{0}(t, x) solve the problem of optimal stabilization and an equality

∫_{t_{0}}^{∞} ω(t, x^{0}[t]; u^{0}[t]) dt = min ∫_{t_{0}}^{∞} ω(t, x[t]; u[t]) dt = V^{0}(t_{0}, x(t_{0}))  (7)

holds.