Time-Optimal Control Problem for a Special Class of Control Systems: Optimal Controls and Approximation in the Sense of Time Optimality

We consider the time-optimal control problem to the origin for a class of nonlinear systems, called dual-to-linear systems. We obtain the general description of possible optimal controls. In particular, we show that optimal controls take the values − 1, 0, and + 1 only and have a ﬁnite number of points of discontinuity. We describe a class of nonlinear afﬁne control systems which can be approximated by dual-to-linear systems in the sense of time optimality.

connection with the controllability and stabilizability problems for nonlinear systems. Later on, the class of triangular systems was considered in many works, in particular, in connection with the problem of linearizability, i.e., the possibility to transform a nonlinear system to a linear one. A way using "Lie brackets technique" was proposed in 1973 by Krener [2] and developed in numerous works. However, the class of nonlinear linearizable systems is rather small. So, the next step was to develop methods of approximation (in some sense) of a given nonlinear system by a linear one. In [3] the approximation was considered for nonlinear affine systems with real analytic right-hand side, and a concept of approximation in the sense of time optimality was introduced. Moreover, necessary and sufficient conditions were obtained, under which the system is approximated by a certain linear system.
If these conditions are not satisfied, the question arises, how to approximate the original system by another nonlinear affine system of a simpler form. Further progress was achieved by developing the algebraic approach [4][5][6][7][8][9][10]. As a result, it was shown that the approximating system can be constructed with the use of some special structures in the algebra of nonlinear power moments.
In the present paper, we consider the time-optimal control problem for affine systems with real analytic right-hand side including control and the first coordinate only. It turns out that optimal controls take values −1, 0, +1 and have a finite number of points of discontinuity. We study the question of approximation in the sense of time optimality, following the approach proposed in [3,5]. We find conditions under which a system of the considered class approximates an affine control system. These conditions are "dual" to the corresponding conditions for systems approximated by linear ones [3]. That gives a certain reason to interpret such systems as dual-to-linear systems.
The paper is organized as follows. In Sect. 2, we consider the time-optimal control problem for dual-to-linear systems and show that optimal controls take values −1, 0, and +1 only and have a finite number of points of discontinuity, and describe possible optimal controls. Section 3 contains the three-dimensional example. Finally, in Sect. 4 we describe the structure of right ideals induced by dual-to-linear systems in the algebra of nonlinear power moments and consider the question of approximation in the sense of time optimality.
Lemmas 2.1 and 2.3 lead to the following theorem.
Our next goal is to estimate the possible number of switching points and specify the character of optimal trajectories.

Lemma 2.4 If t is a switching point of the optimal control u(t), then x
, and c is a multiple root of the function P(z).
Proof The first statement follows from (8) and (7). Let us turn to the second statement.
Let us explain the meaning of Lemma 2.4. Suppose that z 1 , . . . , z m ∈ [−α, α] are multiple roots of P(z), and z m+1 , . . . , z p ∈ [−α, α] are simple roots. Let us consider the graph of the function P(z) and suppose that the point z = x 1 (t) moves along the axis z when t runs through the time interval [0, θ ]. Relation (7) means that x 1 (t) belongs to the connected component of the set {z : P(z) ≥ 0} containing the point z = 0. Moreover, x 1 (t) can change the direction of its movement at the points z 1 , . . . , z p only, and x 1 (t) can "stay" at the points z 1 , . . . , z m only.
Below, we denote by , be a continuous piecewise linear function and the points 0 We say that the function ϕ(t) We note that ϕ(t) is continuous and piecewise linear.
, be a piecewise constant control steering the initial point x 0 to the origin in the time θ , and x(t) be the corresponding trajectory. Let the points 0 < τ 1 < τ 2 < τ 3 < θ be such that . Let the function x 1 (t) be obtained from x 1 (t) by a transposition w.r.t. the points τ 1 , τ 2 , τ 3 , and u(t) =˙ x 1 (t). Then we say that the control u(t) is obtained from u(t) by an admissible transposition.
The next lemma follows from the form of the system (1).

Lemma 2.5 Let u(t), t ∈ [0, θ], be a control that steers the point x 0 to the origin in the time θ . Suppose the control u(t) is obtained from u(t) by an admissible transposition;
then u(t) steers x 0 to the origin in the same time θ .
, be a continuous piecewise linear function and η ∈ R. We say that the function ϕ(t) takes the value η k times iff the pre-image ϕ −1 (η) ⊂ [0, θ] has k connected components (points or segments).

Lemma 2.6
Suppose a continuous piecewise linear function ϕ(t), t ∈ [0, θ], such that ϕ(θ) = 0, takes a certain nonzero value at least three times; then, for any k ≥ 1 there exists a function ϕ(t), which is obtained from ϕ(t) by a finite number of transpositions and has at least k pairwise different local extreme values.
Proof By supposition, for a certain number η = 0 the pre-image ϕ −1 (η) has at least three connected components and ϕ(θ) = 0. Then, there exist three disjoint is nonzero on each of them. Hence, it is the same at least on two of these intervals. Let us assume that sign(ϕ(t) − η) = 1 for t ∈]t 1 , t 2 [∪]t 5 , θ[ (the other cases are considered analogously). Then there exist a nonzero number η > η and three points τ 1 , Moreover, η can be chosen so that it is not a local extreme value of the function ϕ(t).
Let us denote by ϕ(t) the function which is obtained from ϕ(t) by a transposition w.r.t. the points τ 1 , hence, τ 1 is a local maximum point of the function ϕ(t). Analogously, τ 3 is a local minimum point of ϕ(t). Hence, η is a local extreme value of ϕ(t). Moreover, all local extreme values of ϕ(t) (if any) are also local extreme values of ϕ(t) (maybe, corresponding to other extreme points). Hence, ϕ(t) has at least one local extreme value more than ϕ(t).
We note that ϕ(t) takes the (nonzero) value η at least three times, so, we may repeat the described procedure, applying it to ϕ(t). Applying this procedure k times, we obtain a function described in the statement of the lemma.
Proof Assume the converse. Then there exists a sequence . . , f n (z) are holomorphic; then F k (z) and F(z) are also holomorphic. Then, F(z) has a finite number of roots in D. Due to Rouché's Theorem for complex-valued functions, there exists a number r such that the functions F(z) and F k (z) have the same number of roots in D for k ≥ r ; this contradicts the assumption. Proof Assuming the converse, we suppose θ is the optimal time, u(t) is an optimal control, and x(t) is an optimal trajectory. Let N be a constant from Corollary 2.1. Applying Lemma 2.6 to x 1 (t), we obtain a function x 1 (t) having at least N + 1 are switching points for u(t). Lemma 2.5 implies that u(t) steers x 0 to the origin in the optimal time θ , hence, u(t) is optimal. This contradicts Corollary 2.1.
Roughly speaking, the previous results mean that the first coordinate of the optimal trajectory, x 1 (t), does not take the same value more than twice; the unique exception is the case x 0 1 = 0, when the value x 1 (t) = 0 can be taken three times. In particular, this implies that x 1 (t) has no more than one strong local maximum and one strong local minimum on the interval ]0, θ [ (and each of these values can be taken only once). Note that all "zero pieces" of x 1 (t) (corresponding to the intervals where u(t) = 0) can be put between these extreme points by admissible transpositions.
Thus, let us describe possible optimal controls. We introduce the following notation for three constant functions where k ≥ 1 and Moreover, any other optimal control can be reduced to this form by a finite number of admissible transpositions.   points t i of an optimal control can be obtained from the fact that the numbers x 1 (t i ) are roots of a function P(z) = −ψ 0 − n i=2 ψ i P i (z) for some ψ 0 , ψ 2 , . . . , ψ n . In next section, we use this observation to specify the possible optimal controls for a concrete example.

Example
Consider the time-optimal control problem of the forṁ The characteristic function of this system has the form P(z) = −ψ 0 − ψ 1 z − ψ 3 z 3 .
Hence, it has no more than three simple different real roots or one multiple root and no more than one simple root. Suppose x 0 1 > 0 and consider "typical" cases for P(z) when ψ 3 = 0 (Fig. 1).
Let us ignore the controls having no more than one switching point (they obviously are optimal, since they are optimal for the linear sub-systemẋ 1 = u,ẋ 2 = x 1 ). Then we get the following possible forms of optimal controls (Figs. 2, 3, 4).
This initial point can be also steered to the origin by the bang-bang control with two switchings u(t) = ( p a 1 • m b 1 • p a 2 )(t), where a 1 = 0.94, b 1 = 3.60, a 2 = 1.66. However, this control is not optimal, since it does not belong to the cases (i)-(iv) described above. In fact, we have x 1 (a 1 ) = 1.94 and x 1 (a 1 + b 1 ) = −1.66, and it is easy to check that x 1 (a 1 ) < 2|x 1 (a 1 + b 1 )| and x 1 (a 1 ) > 1 2 |x 1 (a 1 + b 1 )|. For this control, the time of motion equals T = 6.19. The components of the trajectory are given in Fig. 5 (right picture).

Approximation in the Sense of Time Optimality
In this section, we describe the class of affine control systems that are equivalent to the ones of the form (1) in the sense of time optimality. Let us consider the time-optimal control problem for affine control systems of the form where a(t, x), b(t, x) are real analytic on a neighborhood of the origin in R n+1 . We introduce the operator S a,b (θ, u) that maps a pair (θ, u) to the initial point x 0 , which is steered to the origin by the control u = u(t) in the time θ , i.e., S a,b (θ, u) = x 0 . This operator admits the following series expansion [3,5]: where ξ m 1 ...m k (θ, u) are nonlinear power moments of the function u(t), and v m 1 ...m k are constant vector coefficients defined via a(t, x) and b(t, x) by the following formulas. We introduce the operators R a and R b acting as for any real analytic vector function f (t, x). We use the notation ad 0 where E(x) ≡ x. Now, we recall some concepts and results concerning the application of the free algebras technique proposed and developed in [4][5][6][7][8][9][10]. Different approaches based on series representations close to (12) can be found in [13][14][15][16][17][18][19]. We consider nonlinear power moments ξ m 1 ···m k (θ, u) as words generated by the letters ξ i (θ, u), i.e., assume that the word ξ m 1 . ..m k (θ, u) is a concatenation of the letters ξ m 1 (θ, u), . . . , ξ m k (θ, u). Then the linear span of nonlinear power moments becomes an associative non-commutative algebra. It can be shown that nonlinear power moments are linearly independent as functionals on L ∞ [0, θ] (for θ > 0); therefore, the above-mentioned algebra is free. Hence, it is isomorphic to an abstract free algebra generated by abstract elements {ξ i } ∞ i=0 (over R) with the multiplication ξ m 1 ···m k := ξ m 1 ∨ · · · ∨ ξ m k . We denote this algebra by A and call it the algebra of nonlinear power moments. We introduce the free Lie algebra L generated by with the Lie bracket operation [ 1 , 2 ] := 1 ∨ 2 − 2 ∨ 1 , 1 , 2 ∈ L. Then A is a universal enveloping algebra for L. Finally, we introduce the inner product ·, · such that {ξ m 1 ...m k : k ≥ 1, m 1 , . . . , m k ≥ 0} is an orthonormal basis of A. We note that the algebra A admits the natural grading, We note that the series (12) defines the linear mapping v : A → R n by the rule v(ξ m 1 ...m k ) = v m 1 ...m k . Thus, one can consider an abstract analog of the series (12), i.e., the series of elements of A with constant vector coefficients of the form Below, we assume that v satisfies the Rashevsky-Chow condition [20,21] v(L) = R n .
We recall that (15) is an accessibility condition for the system (10), i.e., it guarantees that the set of those x 0 , that can be steered to the origin, has a nonempty interior, and the origin belongs to the closure of this interior. For a given system (10), we consider its core Lie subalgebra L a,b defined by We introduce also the right ideal J a,b := Lin{ ∨ z : ∈ L a,b , z ∈ A +R} and denote by J ⊥ a,b the orthogonal complement of J a,b in A. Let the Rashevsky-Chow condition (15) be satisfied; then L a,b is of codimension n. Suppose 1 , . . . , n ∈ L are such that and i ∈ A w i , i = 1, . . . , n. For any element a ∈ A, we denote by a the orthoprojection of a on the subspace J ⊥ a,b . One can show [5] that there exists a nonsingular analytic transformation z = (x) of a neighborhood of the origin such that and ρ i ∈ ∞ j=w i +1 A j , i = 1, . . . , n. In other words, ( 1 , . . . , n ) is the main part of the series S a,b . One can show that there exists a systeṁ x = a * (t, x) + ub * (t, x), a * (t, 0) ≡ 0, (18) such that Moreover, one can achieve a * (t, x) ≡ 0. Such a system can be interpreted as an algebraic approximation of the initial system. Let us consider the time-optimal control probleṁ where (20) is an algebraic approximation of (10). The question is whether the timeoptimal control problem (20), (21) approximates the initial time-optimal control problem (10), (11). Below, we recall the corresponding definition and result [5].
We suppose ⊂ R n \{0}, 0 ∈ , is an open domain such that, for any x 0 ∈ , there exists the unique solution (θ * x 0 , u * x 0 ) of (20), (21). By U a,b x 0 (θ ) we denote the set of all admissible controls which transfer the point x 0 to the origin by virtue of the system (10) in the time θ , and by θ x 0 we denote the optimal time for (10), (11). Then θ x 0 = min{θ : U a,b x 0 (θ ) = ∅}.

Definition 4.1
We say that the nonlinear time-optimal control problem (20), (21) approximates the time-optimal control problem (10), (11) in the domain iff there exists a nonsingular real analytic transformation of a neighborhood of the origin, (0) = 0, and a set of pairs ( θ x 0 , u x 0 ), x 0 ∈ , such that u x 0 ∈ U a,b (x 0 ) ( θ x 0 ) and Now we recall the main result of [5].
Theorem 4.1 Let the system (10) satisfy the Rashevsky-Chow condition (15). We assume that the elements 1 , . . . , n are chosen by (16) and consider the system (20), whose series has the form (19). Let us suppose that there exists an open domain ⊂ R n \{0}, 0 ∈ , such that (i) the time-optimal control problem (20), (21) has a unique solution (θ * x 0 , u * x 0 ) for any x 0 ∈ ; (ii) the function θ * x 0 is continuous for x 0 ∈ ; (iii) when considering the set K = {u * x 0 (tθ * x 0 ) : x 0 ∈ } as a set in the space L 2 (0, 1), the weak convergence of a sequence of elements from K implies the strong convergence.
Moreover, if the set K = {u x 0 (tθ x 0 ) : x 0 ∈ } also satisfies condition (iii), where (θ x 0 , u x 0 ) is a solution of the time-optimal control problem (10), (11), then in (22) We consider the condition (iii) in a separate way. It is, obviously, satisfied if the system (20) is linear, because in this case K includes only piecewise constant functions having no more than n − 1 switchings. However, in the general case this is an open question about a class of systems satisfying this condition [6]. The results of Sect. 2 allow us to conclude that systems of the form (1) satisfy condition (iii) automatically. Let us describe the form of the approximating series (19) for systems (1) more specifically and obtain the corresponding corollary of Theorem 4.1. We denote D m (x 1 ) := (0, P (m) We denote ξ k 0 := ξ 0 ∨ · · · ∨ ξ 0 (k times). Then Since the functions P 2 (x 1 ), . . . , P n (x 1 ) are linearly independent, the Rashevsky-Chow condition (15) holds. Suppose q 2 < · · · < q n are the indices of the first n − 1 linearly independent elements in the sequence {D k (0)} ∞ k=1 . Then, there exists a system (18) of the forṁ such that J a,b = J a * ,b * . Therefore, (24) is an algebraic approximation of (1). In general, we say that a system of the form (10) is essentially dual-to-linear iff there exists a dual-to-linear system (1) whose ideal coincides with the ideal of (10). The discussion above implies that a system of the form (10) is essentially dual-to-linear if and only if and where m = m 1 + · · · + m k + k, k ≥ 1, m 1 + · · · + m k ≥ 2.

Corollary 4.1 Suppose a system of the form
, j ≥ 1. We suppose that there exists an open domain ⊂ R n \{0}, 0 ∈ , such that (i) the time-optimal control problem for the system (24) has a unique solution (θ * x 0 , u * x 0 ) for any x 0 ∈ ; (ii) the function θ * x 0 is continuous for x 0 ∈ ; Then, for any δ > 0 there exists a domain δ ⊂ R n such that 0 ∈ δ and = ∪ δ>0 δ , and the time-optimal control problem (20), (21) approximates the time-optimal control problem (10), (11) in any domain δ .
Moreover, in the case when the initial system is of the form (1), one can choose θ x 0 = θ (x 0 ) and u x 0 (t) = u (x 0 ) (t). Finally, if q 2 , . . . , q n are odd, then condition (ii) holds automatically. Now, we are ready to explain, why systems (1) are called "dual-to-linear." In [3], we considered systems of the form (10), which are approximated, in the sense of time optimality, by linear systemsẋ = A(t)x +ub(t). We found out that the result analogous to Corollary 4.1 holds, where, instead of (26), the following condition appears where m = m 1 + · · · + m k + k, k ≥ 2, m 1 , . . . , m k ≥ 0.
We note that (26) is obtained as a result of partial replacing of ad j R a R b by ad j R b R a in (27). Such a "duality" of conditions (26) and (27) justifies our term "dual-to-linear systems."

Conclusions
In the paper, we have considered one special class of nonlinear control systems, called dual-to-linear systems. For these systems, we studied the time-optimal control problem and gave the explicit description of the possible character of optimal controls. The three-dimensional example was given to illustrate these results. We described the class of nonlinear affine control systems, which can be approximated by dual-tolinear systems in the sense of time optimality. Our analysis gave reason to conclude that dual-to-linear systems are, in a certain sense, close to linear systems.
It is well known that the time-optimal control problem for a linear system admits an interpretation in terms of the Markov moment problem [23,24]. The results of the present paper open the following direction of the further research: find a moment interpretation for dual-to-linear systems. The explicit form of optimal controls can be considered as a starting point of such an investigation.