Controllability for Problems with Mixed Constraints

For a controlled system with mixed equality- and inequality-type constraints and a geometric constraint that is a nonempty closed convex set, sufficient conditions for the existence of feasible positional controls are obtained in terms of first derivatives of mappings that define the mixed constraints. In addition, in terms of the first and second derivatives of these mappings, sufficient conditions for the existence of feasible positional controls are found that are also applicable in the case of degeneration of the first derivatives of the mappings.


INTRODUCTION
For the controlled systemẋ = F (x, u, t), x(t 0 ) = x 0 , let mixed and geometric constraints have, respectively, the form and u(x, t) ∈ U.
Here x ∈ R n is the state variable, u ∈ R m is the control vector, t ≥ t 0 is time, the mapping F : R n × R m × R → R n is given and is continuous in the first and second variables, measurable in the third variable, and bounded in a neighborhood of the fixed point (x 0 , u 0 , t 0 ), and the given set U ⊂ R m is nonempty, closed, and convex. The vector functions G l : R n × R m × R → R s l , l = 1, 2, are continuous; relations (2) are satisfied coordinatewise. By a control we mean a continuous function u of the variables x and t defined in some neighborhood of the point (t 0 , x 0 ) and such that u(x, t) ∈ U . We say that system (1)- (3) is locally solvable at a point (x 0 , u 0 , t 0 ) if there exists a neighborhood Ω of the point (x 0 , t 0 ), a continuous mappingū : Ω → U , a number τ > 0, and an absolutely continuous functionx : [t 0 , t 0 + τ ) → R n such thatū(x 0 , t 0 ) = u 0 , for all t ∈ [t 0 , t 0 + τ ) one has the relations G 1 x(t),ū x(t), t , t = 0, G 2 x(t),ū x(t), t , t ≤ 0, and the functionx(·) is a solution of the Cauchy probleṁ i.e.,ẋ(t) = F (x(t),ū(x(t), t), t) for a.a. t ∈ [t 0 , t 0 + τ ) andx(t 0 ) = x 0 . The indicated functionū(·) is conventionally called an feasible positional control and the pair (x(·),ū(·)) is called a feasible process.
Our goal is to obtain sufficient local solvability conditions for system (1)-(3) both in terms of the first derivative of the mappings G 1 and G 2 and in terms of the second derivative of these mappings.

FIRST-ORDER REGULARITY
In this section, we give conditions for the regularity of mixed constraints in terms of the first derivative. Assume that By I we denote the set of all those indices i ∈ {1, . . . , s 2 } for which the ith component G 2,i (x 0 , u 0 , t 0 ) of the vector G 2 (x 0 , u 0 , t 0 ) is zero. For a vector function G l , l = 1, 2, by G l u we denote the matrix of its first partial derivatives with respect to the components of the vector u, and for an arbi- . Theorem 1. Let the mappings F and G l , l = 1, 2, be continuous, let the mappings G l be strictly differentiable with respect to u at the point (x 0 , u 0 , t 0 ) uniformly in x, and let the following assumptions be true: Then system (1)- (3) is locally solvable at the point (x 0 , u 0 , t 0 ).
In the case of G 2 (x 0 , u 0 , t 0 ) ̸ = 0, the proof is carried out in a similar way with the replacement of G 2 by the mapping obtained by crossing out the ith components G 2,i , i ̸ ∈ I, of the vector function G 2 . The proof of the theorem is complete.

Remark 1.
If there are no equality-type mixed constraints in problem (1)-(3), then assumptions (R1) and (R2) take the form Note also that a sufficient (but not necessary) condition for (R2 ′ ) is the condition of linear independence of the vectors (G 2 u (x 0 , u 0 , t 0 )) i , i ∈ I. Consider the controlled system (1), (2) with the control This system is said to be locally solvable at a point (t 0 , x 0 ) if there exists a τ > 0, an absolutely continuous function x : [t 0 , t 0 + τ ) → R n , and a measurable essentially bounded function u : for a.a. t and the function x = x(·) is a solution of the Cauchy probleṁ The function u(·) is conventionally called a feasible program control . The local solvability of system (1)-(3) at a point (x 0 , u 0 , t 0 ) implies that there exists a feasible program control u(·) of system (1), (2), (11). Indeed, ifū(·) is a feasible positional control andx(·) is the corresponding trajectory, then u(t) ≡ū(t,x(t)) is a feasible program control. The converse statement, generally speaking, is not true. Let us illustrate the above with an example. i.e., n = m = 2, G 1 (x, u, t) = Ψ(u) − x, and G(x, u, t) ≡ 0. This system has the feasible program control u(t) ≡ 0. However, there is no feasible positional controlū for the system under consideration. Indeed, if some functionū is a feasible positional control, then it is continuous,ū(0, 0) = 0, and Ψ(ū(x, t)) ≡ x; i.e.,ū( · , 0) is a continuous right inverse of Ψ in a neighborhood of zero. The latter contradicts Example 2 in [3], where it was shown that there exists no continuous right inverse mapping of Ψ in a neighborhood of zero.
In conclusion, we note that the controlled system in Example 1 satisfies the assumptions of the theorem on the existence of feasible program controls [4,Theorem 2]. The assumptions in [4,Theorem 2] are weaker than those in Theorem 1. Example 1 shows that under the assumptions in [4, Theorem 2] system (1)-(3) may fail to have a feasible program control in a neighborhood of a given point.

SECOND-ORDER REGULARITY
Let us study the solvability of the controlled system (1)-(3) for the case in which conditions (R1) and (R2) are violated. Set uu , l = 1, 2, we denote the matrix of second partial derivatives of the vector function G l with respect to the components of the vector u.
Theorem 2. Let the mappings F and G l , l = 1, 2, be continuous, and let the mappings G l , l = 1, 2, be twice continuously differentiable with respect to u in a neighborhood of the point (x 0 , u 0 , t 0 ). Let the following assumptions be satisfied: Then the system is locally solvable at the point (x 0 , u 0 , t 0 ). Proof. We assume that G 2 (x 0 , u 0 , t 0 ) = 0. If G 2 (x 0 , u 0 , t 0 ) ̸ = 0, then the proof can be carried out in a similar way but with the replacement of G 2 by the vector function obtained by crossing out the ith components G 2,i , i ̸ ∈ I, of the vector function G 2 .
I. For arbitrary x ∈ R n , v = (λ, ν, y) T ∈ (0, +∞) × R m × R s2 , and t ∈ R, we define the function G(x, t, v) by equality (5). Let A l , l = 1, 2, be the same matrices as in the proof of Theorem 1, and let Q l := G l uu (x 0 , u 0 , t 0 ), l = 1, 2. Define the set K by formula (6). In the proof of Theorem 1, it is shown that K is a convex closed cone.
Let h be the vector in assumption (R3). Since −Q 1 [h, h] ∈ A 1 cone (U − {u 0 }), it follows that there exists a number γ > 0 and a vectorν ∈ U − {u 0 } such that By assumption (R4), there exists a vector ξ ∈ U − {u 0 } such that Consequently, there exist numbers θ > 0 and τ ∈ (0, 1) and a vector y ∈ R s2 + such that As above, set v 0 := (1, 0, 0) T ∈ R × R m × R s2 and define sets K and C by equalities (7). Then we have the representations (8) and (9) and II. Let us show that the mapping G is 2-regular at the point (x 0 , t 0 , v 0 ) with respect to the cone K in the direction of the vector h 0 := (θ, h, −A 2 h) T , i.e., that The inclusion h 0 ∈ K follows from (9) and the definitions of the set K and the vector h 0 . Equality (8) and assumption (R3) imply that Let us prove the inclusion (17). We have Therefore, Here the second equality follows from the relations (14), A 1 ξ = 0, and (16). The inclusion (17) has been proved.
In conclusion, let us give a simple example illustrating the fact that the regularity conditions for (R3) and (R4) are essential. Consider the controlled systeṁ x = u, x(0) = 0, u 2 + t 2 ≤ 0.
It is obvious that this system is not solvable at the point (0, 0, 0) and assumptions (R3) and (R4) are violated for this system.

OPEN ACCESS
This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.