An Estimation from within of the Reachable Set of Generalized Nonlinear R. Brockett Integrator with Small Vector Nonlinearity

open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper the generalized nonlinear R. Brockett integrator with small vector nonlinear addition to the rigth-hand side of the corresponding differential equations is considered. More precisely, we investigate the possibility to estimate from within the corresponding reachable set. We obtain some ellipsoidal estimation from within in an efficient form.


Introduction
One of the fundamental problems in the Control Theory of dynamic systems is the problem of estimation of reachable sets.Reachable sets of controlled objets are the subject of many studies in Mathematical Control Theory.They are studied from different points of view, for example, estimate from above, approximate in the sense of the Hausdorff metric simply constructed convex compact,...In this paper, we obtain a non-trivial estimation from within for the reachable set of the generalized nonlinear R. Brockett integrator with small vector nonlinearity.Such estimations are object of interest for the Optimal Control Theory and its Applications.

Problem formulation
We consider the generalized nonlinear R. Brockett integrator with small vector nonlinearity of the form with zero initial condition x(0) = 0. ( Where x ∈ R 3 , ε 0, g i (x, u), i = 1, 2, 3 are continuous functions on R 3 × U. We impose on the control vector u the constrain We assume that: where c > 0 is a constant, |x| means the norm of the vector x, •,• means the scalar product of vectors and g(x, u) is a continuous vector function with components g i , i = 1, 2, 3. 2. The functions g i (x, u), i = 1, 2, 3 satisfy the local Lipschitz condition in the variables (x, u) on R 3 × U and Note that if ε = 0 the system (1) describes the dynamics of R. Brockett integrator (see [1]- [3]).
Further, we fix the constant T > 0. We consider all possible measurable controls u(t) ∈ U, t ∈ Δ, where Δ = [0, T ], and the corresponding absolutely continuous solutions x(t, u(•), ε) of the system (1) with the initial condition (2).The reachable set we are interested in is where u(•) is an arbitrary measurable control, satisfying (3) for t ∈ Δ.From the general theory of nonlinear controlled objects (see, for example, [4, p. 264]), it follows that D(T , ε) is a compact set in R 3 , where line means the closing of the set.Our objective is to construct for D(T , ε) such a convex compact set K ⊂ R 3 , so that for sufficiently small ε 0 where Int K means the interior of the set K. We denote by L ∞ (Δ) (see [5, p. 31 For any measurable functions u(t) ∈ U, t ∈ Δ (see ( 3)) the corresponding solution x(t, u(•), ε) is uniquely defined with the initial condition (2).Note that it is possible to consider the admissible controls as the elements of L ∞ (Δ).
We will estimate the reachable set D(T , ε) of the system (1), (2) from within.
Note that when ε = 0 the corresponding estimate is obtained in the work [3] of M. S. Nikolskii.

Solution of problem
These are the main results of the paper.We denote by y(T , u(•)) the solution of R. Brockett system of equations (see [1]- [3]) where y(0) = 0 and the admissible u(t) ∈ U, t ∈ Δ.
Then it is easy to show that the solution x(T , u(•), ε) of the system (1), (2) allows the following representation for admissible u(t) ∈ U, t ∈ Δ: where and A is a linear vector-bounded operator, acting from L ∞ (Δ) to R 3 by the formulas for its component of kind ] is a continuous vector quadratic form (see [5], [6]) with components ) R(u(•), ε) is a nonlinear vector function, and also (see (1), ( 5)) Using the inequality (4) and the results of [7] we can show that the function where Using the local Lipschitz condition in (x, u) of the functions g i (x, u), i = 1, 2, 3 on R 3 × U, the inequality (16), we can prove the continuity in (u(•), ε) of the nonlinear mapping R(u(•), ε) on the set of all admissible controls u( From formula (11) we see that, AL ∞ = R 3 , i. e. the linear operator does not realize a covering of the space R 3 .We consider the admissible control ũ(t) ∈ U, t ∈ Δ, with the components where It is easy to see that (see ( 11)-( 14)) for ũ(t), t ∈ Δ, and its corresponding solution x(t) of system (1), ( 2) the equalities are realized.We consider for t ∈ Δ and μ ∈ (0, 1] the admissible control where the measurable function ν(t) ∈ R 2 satisfies on Δ the inequality From ( 9), ( 10), ( 19) and ( 20) we obtain the following formula for the solution x μ (t) of the system (1), (2), that corresponding to the control u μ (•) : where C[ξ, η] is a continuous symmetric bilinear form (here ξ, η are arbitrary elements of L ∞ (Δ)).We note that (see ( 12)-( 14), ( 17)-( 19)) for arbitrary ξ, η from L ∞ (Δ) and ) where ν i (t), i = 1, 2, are the components of the vector function ν(t).Further (see (18)), for r ∈ Δ.
Changing the order of integration in the first iterated integral (24) and using the formula (25), we can rewrite (24) in the following form: We consider now the linear operator N, acting from L ∞ (Δ) to R 3 by the formula We note that the linear operator N is closely connected with the formula (22).
is realized.