Abstract
The attainable sets of the nonlinear control systems with integral constraint on the control functions are considered. It is assumed that the behavior of control system is described by differential equation which is nonlinear with respect to phase-state vector and control vector. The admissible control functions are chosen from the closed ball centered at the origin with radiusμ0inLp([t0,θ];ℝm) (p∈(1,+∞)). Precompactness of the solutions set is specified, and dependence of the attainable sets on the initial conditions and on the parameters of the control system is studied.
Highlights
Control problems with integral constraints on control arise in various problems of mathematical modeling
The following proposition asserts that the set of solutions and the attainable sets of the control system 1.1 with constraint 1.2 are bounded
In 19, 20, the example is given which illustrates that the attainable set of nonlinear control system with geometric constraint on control is not closed
Summary
Control problems with integral constraints on control arise in various problems of mathematical modeling. In 4–10 , topological properties and numerical construction methods of the attainable sets of linear control systems with integral constraint on control functions are investigated. Approximation method for the construction of attainable sets of affine control systems with integral constraints on the control is given in 11, 13. It is obvious that the set of all admissible control functions Upμ[0] is the closed ball centered at the origin with the radius μ0 in Lp t0, θ ; Rm. It is assumed that the right-hand side of the system 1.1 satisfies the following conditions. Note that the conditions a – c guarantee the existence, uniqueness, and extendability of the solutions up to the instant of time θ for every given u∗ · ∈ Upμ[0] and x0 ∈ X0. HC U, V denotes the Hausdorff distance between the sets U ⊂ C t0, θ ; Rn and V ⊂ C t0, θ ; Rn
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