Local uncontrollability for affine control systems with jumps†

ABSTRACT This paper investigates affine control systems with jumps for which the ideal If(g1, …, gm) generated by the drift vector field f in the Lie algebra L(f, g1, …, gm) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.


Introduction
Algebraic-geometric methods are successfully applied in control systems theory with relevant results regarding controllability, observability, robust stability, etc. In this paper, the analysis is focused on two special features involving nonlinear affine control systems with jumps associated with uncontrollability. The main theoretical ingredients are presented in Vârsan (1999), where the analysis includes as applications, among others, the control systems and their gradient flow representation, without considering any jumps. A special attention is paid to geometric methods applied to affine control systems which, in a way, can be taken as non-holonomic constraints appearing in mechanics and, therefore, directly connected with hyperbolic differential equations analysed in the first part of Vârsan (1999).
Motivated by the analysis, with remarkable results, performed in Vârsan (1999) regarding gradient flow representations for some solutions, Lie algebras, gradient systems in a Lie algebra, algebraic representation of gradient systems and their integral manifolds, this paper develops and improves more results in the current literature. The techniques and mathematical tools used in this paper (see also Doroftei & Treanţȃ, 2012;Parveen & Akram, 2012;Treanţȃ & Vârsan, 2013b, 2014Vârsan, 1994) lead us to a nice gradient flow representation of solutions satisfying the affine control system with jumps considered in the present work.
The standard form of a control system includes trajectories without any jumps. In our non-standard model, we consider piecewise constant control functions. It induces CONTACT Savin Treanţȃ savin_treanta@yahoo.com, savin.treanta@upb.ro † Dedicated to the memory of Constantin Vârsan a finite set of jumps for each trajectory and the integral form of a trajectory uses new bounded variation controls as integration variables. In other words, the analysis of non-standard control systems includes a finite set of jumps which influence the trajectories of the control systems. Any affine control system with jumps has the property that its trajectories are contained in the kernel of a linear first-order partial differential equation (PDE). It should be noted that in our case the trajectories are piecewise smooth bounded variation functions. This explains the non-standard writing used in the present work. The main result can be easily connected with so-called unreachability of an affine control system. When controllability or uncontrollability properties are involved, we need a special analysis and our paper presents the details of the uncontrollability properties connected with complete affine control systems under jumps conditions. The purpose of this paper is to formulate and prove Lie-algebraic sufficient conditions which involve the local uncontrollability of a given affine control system with jumps. In the main result of this paper, the uncontrollability conditions are not influenced when a non-standard affine control system is replaced by a standard one. In addition, at each fixed time, the range of affine control system is contained in a shifted manifold using the flow of the drift vector field, while in a driftless control system, the range of the control system is contained in the same kdimensional manifold (k < n). It is to be mentioned that, for driftless affine control systems, the k-dimensional differentiable manifold is constructed in Parveen and Akram (2012). Also, the reachable set of affine control systems with jumps, at any time, can be included in the values of a given smooth mapping. This part, under more restrictive assumptions, can be associated with the analysis performed in Vârsan (1994). For other different but related viewpoints on this subject (discrete-time piecewise linear delay systems and continuous-time Markovian jump systems with time-varying delay and deficient transition descriptions), the reader is directed to Qiu, Feng, and Yang (2009) and Qiu, Wei, and Karimi (2015). As well, by using the concepts of graph completion and generalised solution, some impulsive control systems (where the evolution equation depends linearly on the time derivative of the control function) have been also studied in a series of works (see Bressan, 1987;Bressan & Rampazzo, 1988, 1991, 1994. In the case of measurable control functions, various definitions of generalised trajectories have been introduced in the literature by considering suitable limits of classical solutions, under key assumptions involving either the total variation of the control function or the commutativity of the vector fields. A basic assumption in Bressan and Rampazzo (1991) is that the vector fields g α := g α , e α , α = 1, m, commute, i.e. their Lie brackets vanish identically. By the commutativity hypothesis, the authors construct a transformation which reduces the initial control system to a control system in the usual form, with absolutely continuous trajectories. A generalised solution, for the control system studied in Bressan and Rampazzo (1991), is introduced as a trajectory t → x(u, t ) for which there exists a sequence of controls v k ∈ C 1 ([0, T ], R n ) such that v k (0) = u(0) = x, v k → u in the L 1 -norm, and the corresponding trajectories x(v k , ·) have uniformly bounded values and tend to x(u, ·) in the L 1 -norm. The case when the vector fields g i , i = 1, m, do not commute is analysed in Bressan and Rampazzo (1994) by considering a quotient control system. In the present paper, applying the standard Picard's iterative procedure (used for ordinary differential equations (ODEs)), we get a unique solution for our non-standard control system by combining a smooth mapping (an adequate composition of flows) with a solution satisfying an auxiliary affine control system in the space of parameters.
The paper is organised as follows: in Section 2, we present the notations, definitions and the preliminary results to be used in the sequel; in Section 3, a constructive proof for the problem proposed is presented, and Section 4 formulates the conclusion of this work.

Preliminaries
We begin this section by introducing some notations, definitions and preliminary results which will be used further.
Definition 2.1: The space C 1 b (B(x 0 , 2ρ) ⊆ R n ; R) consists of all first-order continuously differentiable functions h(x) : B(x 0 , 2ρ) → R (see B(x 0 , 2ρ) as the ball centred in x 0 ∈ R n and radius 2ρ, with ρ > 0 fixed) for which

Definition 2.3: The set of admissible controls
Consider a finite set of smooth vector fields and associate an affine control system with jumps where is an admissible control v ∈ V C (see Definition 2.3).

Definition 2.4:
We say that T > 0 and C > 0 are sufficiently small, if the following conditions are fulfilled: Applying the standard Picard's iterative procedure (used for ODEs), we get a unique solution satisfying the affine control system (2) Comment 2.1: Let us establish the form of equation given in (2) on each interval of partition (see Definition 2.2). Consider the first interval [0, τ 1 ] in our partition, associated with v ∈ V C (see Definition 2.3), where t = τ 1 is a jump of the control function. In this situation, the equation given in (2) will be written as follows: for each v ∈ V C , provided T > 0 and C > 0 are sufficiently small (see Definition 2.4). Repeat this procedure on each interval [τ j , τ j+1 ], j = 0, N − 1. In addition, the partition associated with x v ∈ PSBV ([0, T ]; R n ) coincides with that of v ∈ V C and the jumps satisfy the algebraic equations Assuming Definition 2.4 and having in mind the Banach fixed-point theorem, the following result can be formulated.
satisfying the affine control system with jumps in (2).
Definition 2.5: The affine control system (2) Definition 2.6: Let g 1 , . . . , g m ⊂ C ∞ (R n ; R n ) be a finite set of smooth vector fields. Define the Lie algebra generated by these vector fields, denoted L = L g 1 , . . . , g m ⊂ C ∞ (R n ; R n ), as being algebra which contains Lie products (of arbitrary length) composed by elements of g 1 , . . . , g m and of their linear span.
Definition 2.7: Let L be a Lie algebra. A Lie ideal of L is a linear subspace I ⊆ L such that [ f ; g] ∈ I, whenever f ∈ L and g ∈ I.
Definition 2.8: The ideal g 1 , . . . , g m ) generated by f in the Lie algebra L( f , g 1 , . . . , g m ) is obtained as the Lie subalgebra generated by an infinite set of smooth vector fields that is,

Definition 2.9:
The ideal I f (g 1 , . . . , g m ) is called locally of finite type if there exists a system of generators {X 1 , . . . , X M } ⊆ I f (g 1 , . . . , g m ) such that any X ∈ I f (g 1 , . . . , g m ) can be written as

Main result
Our main goal is to stipulate Lie-algebraic sufficient conditions leading us to the local uncontrollability of the affine control system with jumps given in (2). More precisely, we shall solve the following: Main problem. Under the following conditions: . . , g m ) generated by f in the Lie algebra L( f , g 1 , . . . , g m ) is locally of finite type (see Definitions 2.8 and 2.9); prove that (2) is locally uncontrollable.
The main result (see Theorem 3.1) can be easily connected with so-called unreachability of an affine control system. This indicates that the reachable set x 0 ), v ∈ VĈ} satisfies int R n R(t, x 0 ) = ∅, for any t ∈ [0,T ], even if the largest set of controls v ∈ VĈ is used. This set of admissible controls, v ∈ VĈ, generates analytic solution in the auxiliary control system, provided the assumptions in Vârsan (1994) (which are more restrictive) will be used.
The algebraic-geometric methods used here lead us to a nice gradient flow representation which is based, this time, on a complete smooth mapping including the flow generated by the drift { f } (see Parveen & Akram, 2012, for driftless affine control systems). On the other hand, the reduced smooth mapping (as a second component of the main mapping) is generated by the corresponding infinitesimal generators in the ideal I f (g 1 , . . . , g m ) and the evolution of the original dynamical system will be described using an auxiliary affine control system with jumps. In addition, the Lie subalgebra I f (g 1 , . . . , g m ), under the hypotheses stipulated in Main problem, can be imbedded as a kernel of a linear first-order PDE. It is to be mentioned that, for driftless affine control systems, the k-dimensional differentiable manifold is constructed in Parveen and Akram (2012), provided dim L(g 1 , . . . , g m )(x 0 ) = k is assumed. In particular, under the singularity assumption, dim I f (g 1 , . . . , g m )(x 0 ) = k < n, we can extend the auxiliary results (see Lemmas 3.1 and 3.2) involving a larger class of admissible controls described by piecewise smooth bounded variation functions (see v ∈ PSBV ([0, T ]; R m )). Taking into account the computation complexity of the derived results, certain calculations will be omitted in our presentation. For more details, there will be indicated bibliographic references.
The algorithm for Main problem relies on the gradient flow representation of the solutions {x v (t, x 0 ) : t ∈ [0,T ], v ∈ VĈ} satisfying the affine control system with jumps in (2). In this respect, fixing a system of generators {g 1 , . . . , g m , h m+1 , . . . , h M } ⊆ I f (g 1 , . . . , g m ), we define an adequate composition of flows: Here are the local solutions satisfying ODEs driven by the vector fields f , g i and h j correspondingly, with the initial conditions In addition,T > 0,â k > 0, k = 1, M, are sufficiently small such that With these minor arrangements, we notice that the smooth mapping in (5) satisfies The main goal of the so-called gradient flow representation in (5) is to make sure that each infinitesimal generator can be represented with respect to the fixed system of generators as follows: a (n × M) matrix whose column j is the vector Z j (y), j = 1, M, and also ad Z {Z 1 , . . . , where the (M × M) smooth matrix A(t, p) is a nonsingular one for any t ∈ [0,T ], p ∈D M , A(0, 0) = I M (for more details, see Vârsan, 1999). Both representations (see (5) and (9)) are helpful for extending our considerations, but the non-singularity of the algebraic equations in (9) allows us to introduce an auxiliary affine control system in the space of parameters Here, the smooth vector fields {q 1 , . . . , q m } ⊆ C ∞ ([0,T ] ×D M ; R M ) are found as the unique solution of the algebraic equations where the non-singular matrix A(t, p) is given in (9) and {e 1 , . . . , e M } ⊆ R M stands for the canonical basis. Notice that 0 <T ≤ T and 0 <Ĉ ≤ C will be taken sufficiently small such that each solution of (10) satisfies (see δ > 0 fixed) for any admissible control v ∈ VĈ. Combining the smooth mapping y = G(t, p; x 0 ) (see (5)) with a solution p v ∈ PSBV ([0,T ]; B(0, δ) ⊆D M ), satisfying the auxiliary control system (10), we get (by a straight computation) as the unique solution of the original affine control system (2), for any v ∈ VĈ.
The last step in our algorithm is to construct a k- (10) and the smooth mappingĜ(p; x 0 ) ∈ B(x 0 , ρ), p ∈D M , is given byĜ Notice that the complete mapping G(t, p; x 0 ) defined in (5) is the composition of the flow F (t ) [z] (generated by the vector field f ) with the reduced smooth map-pingĜ(p; x 0 ) described in (14). In addition, the reduced smooth mappingĜ(p; x 0 ) is generated by a fixed system of generators with respect to which the non-singular algebraic representation (9) is valid. Definition 3.2: A system of generators {X 1 , . . . , X M } for the (locally) of finite-type Lie algebra To get (13), we need to use a (k, x 0 )-minimal system of generators Consequently (see Definition 3.2), are linearly independent, and X j (x 0 ) = 0 for j = k + 1, M. A (k, x 0 )-minimal system of generators, {X 1 , . . . , X k , X k+1 , . . . , X M }, replacing {g 1 , . . . , g m , h m+1 , . . . , h M } in (15), will be found from the original system of generators {g 1 , . . . , g m , h m+1 , . . . , h M } ⊆ I f (g 1 , . . . , g m ) multiplied on the right-hand side by a non-singular (M × M) constant matrix K, i.e.
According to (15), this time, {Ĝ(p; x 0 ) : p ∈D M } in (18) has a reduced form which is more suitable to describe it as a k-dimensional manifold M x 0 ⊆ B(x 0 , ρ). Though the smooth mapping in (5) will be replaced by (18), the final algebraic non-singular representation in (9) will be written according to the original system of generators as follows: where A(t, p) := KÂ(t, p), t ∈ [0,T ], p ∈D M , is the non-singular (M × M) matrix used for constructing the auxiliary control system in (10). The k-dimensional manifold M x 0 ⊆ B(x 0 , ρ) associated with the reduced mappingĜ(p; x 0 ) =Ĝ(p; x 0 ) ∈ B(x 0 , ρ) (see (18) and (20)) will be obtained using the same theoretical ingredients as in Parveen and Akram (2012) (see Lemma 2.1). We shall formulate it as the following lemma: Lemma 3.1: Assume that the conditions (i) and (ii) of Main problem are satisfied and define the reduced mappinĝ (18) and (20)). Then there exist 0 <ρ ≤ ρ and non-constant smooth scalar functions {λ k+1 , . . . , λ n } ⊆ C 1 (B(x 0 ,ρ) ⊆ R n ; R) such that The proof of this lemma is a direct application of the implicit functions theorem. On the other hand, there are some differences between the non-singular algebraic representation in (22) and that which was proved in Lemma 2 of Parveen and Akram (2012). The main difference involves the smooth mapping along which the non-singular algebraic representation in (22) is performed. Here, apart from the smooth mappinĝ G(p; x 0 ), p ∈D M , generated by the (k, x 0 )-minimal system of generators {X 1 , . . . , X k , X k+1 , . . . , X M }, we use the group of diffeomorphisms {F (t ) [z] : t ∈ [−T ,T ], z ∈ B(x 0 , ρ)} obtained as local solutions satisfying ODEs Consider the smooth mappings is generated by the (k, x 0 )-minimal system of generators {X 1 , . . . , X k , X k+1 , . . . , X M } ⊆ I f (g 1 , . . . , g m ) (see (18)). The Lie subalgebra I f (g 1 , . . . , g m ) fulfils the conditions of Lemma 2 in Parveen and Akram (2012) and it allows us to write the following non-singular algebraic representation associated withŷ =Ĝ(p; {∂ t 1 y, ∂ t 2 y, . . . , ∂ t M y} for any t ∈ [0,T ] and p = (t 1 , . . . , t m ) ∈D M .
Then the non-singular algebraic representation in (30) is valid (see A(0, 0) = K). Now, we shall formulate and prove the main result of the present paper. This indicates that each reachable set R(t, x 0 ) is contained in a shifted k-dimensional differentiable manifold (k < n), or int R n R(t, x 0 ) = ∅, provided dim I f (g 1 , . . . , g m )(x 0 ) = k and I f (g 1 , . . . , g m ) is of finite type. This reflects the possibility that the corresponding Lie algebra , v ∈ VĈ satisfies the control system (2) and where the k-dimensional manifold M x 0 ⊆ B(x 0 , ρ) (k < n) is constructed in Lemma 3.1.
Proof: By hypothesis, the conditions of Lemmas 2.1, 3.1 and 3.2 are satisfied such that the smooth mappingŝ y =Ĝ(p; x 0 ) described in Lemma 3.2, can be used to represent the solutions , v ∈ VĈ satisfying affine control system with jumps in (2). Using the non-singular algebraic representation given in Lemma 3.2, associated with the infinitesimal generators ∂ t 1 y, . . . , ∂ t M y , define the smooth vector fields where {e 1 , . . . , e M } ⊆ R M is the canonical basis.

Application to non-holonomic systems
To highlight how the reasonings were designed in this work, we consider the following application. Let us consider a non-holonomic system defined as a nonintegrable system of Pfaff forms (Doroftei, 2001) n j=1 a i j (t, x)dx j + a i0 (t, x)dt = 0, i ∈ {1, 2, . . . , k}, k < n, where the scalar functions a i j (t, x), a i0 (t, x) : I × G → R are of C ∞ -class and I ⊆ R, G ⊆ R n are open sets. Making some assumptions and notations, the foregoing system (31) can be written as follows: where dy = col(dx 1 , . . . , dx k ), with the variable y ∈ R k , k < n, as function depending on (t, x k+1 , . . . , x n ) := τ := (τ 0 , . . . , τ m ) ∈D m+1 := m j=0 (τ 0 j − a j , τ 0 j + a j ), with jumps for which the ideal I f (g 1 , . . . , g m ), generated by the drift { f } in the Lie algebra L( f , g 1 , . . . , g m ), fulfils a degeneracy condition dim I f (g 1 , . . . , g m )(x 0 ) = k < n, for some x 0 ∈ R n . It led us to uncontrollable affine control systems with jumps, where the main (complete) smooth mapping includes the flow generated by the drift { f } combined with a reduced smooth mapping determined by a system of generators in the ideal I f (g 1 , . . . , g m ) ⊆ L ( f , g 1 , . . . , g m ). This reduced mapping is a finite composition of flows starting from a fixed point x 0 ∈ R n and its values can be restricted to a k-dimensional differentiable manifold (k < n) provided we assume dim I f (g 1 , . . . , g m )(x 0 ) = k. The main result (see Theorem 3.1) indicates that each reachable set R(t, x 0 ) is contained in a shifted k-dimensional differentiable manifold (k < n), or int R n R(t, x 0 ) = ∅, provided dim I f (g 1 , . . . , g m )(x 0 ) = k and I f (g 1 , . . . , g m ) is of finite type. For other different but related viewpoints on this subject, the reader is directed to Akram and Lomadze (2009), Jurdjevic and Sussmann (1972), Mailybaev (2003), Polderman and Willems (1998), Pomet (1999), Qiu et al. (2009), Qiu et al. (2015, Sontag (1990), Sussmann (1983), Treanţȃ and Udrişte (2013a) and Treanţȃ (2014).