THE LIMITS OF APPLICABILITY OF THE LINEARIZATION METHOD IN CALCULATING SMALL–TIME REACHABLE SETS1

The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the L2-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets. The conditions depend on the behavior of the controllability Gramian of the linearized system — the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero. The indicated asymptotic behavior occurs for a reasonably wide class of secondorder nonlinear control systems but can be violated for systems of higher dimension. The results of numerical simulation illustrate the theoretical conclusions of the paper.


Introduction
The paper explores the properties of reachable sets of control-affine nonlinear systems with integral constraints over small time intervals. The geometric structure of reachable sets plays an important role in control theory, in particular, in solving problems of control synthesis. Smalltime reachable sets under pointwise (geometric) constraints on control were studied by C. Lobry, H. Sussmann, A. J. Krener, H. Schattler, and C. I. Byrnes (see, for example, [12,19]). In general, the reachable sets of nonlinear systems are not convex and may have a quite complicated structure [1, 3, 11, 13-15, 20, 21]. When some of the parameters of a control system are small (initial deviations from the equilibrium position, disturbances at the input of the system, etc.), the behavior of the system can often be judged by the action of its linear approximation. Here we find out under what conditions this linearization approach is applicable when constructing reachable sets on small time intervals. Will these sets be close to reachable sets of a linearized system? In this paper, we study reachable sets for control-affine systems on small time intervals with integral quadratic constraints on the controls. Reachable sets of nonlinear systems with integral constraints were studied in [5][6][7]16]. If a system is linear, its reachable set is an ellipsoid in the state space. Therefore, an ellipsoid is the reachable set of a linearized system. To establish the proximity of the reachable sets of original and linearized systems, it is necessary first to find out in which cases the reachable set of the original nonlinear system is convex. B. Polyak [17] proved that a nonlinear image of a small ball in a Hilbert space is convex under some regularity assumptions on the mapping. Using this result, he showed that reachable sets of a nonlinear control system are convex if constraints on the control are given by a ball of a sufficiently small radius in L 2 and the linearized system is controllable [16]. Using a time change, we reduce the problem of constructing the reachable set of a system on a small time interval to a similar problem on a unit interval. With this replacement, the integral constraints are given by a ball of small radius, and we apply Theorem 1 from [17] to propose sufficient conditions for the convexity of small-time reachable sets. The application of these conditions requires a study of the asymptotic behavior of the controllability Gramian of the linearized system depending on a small parameter.
Another question is how to evaluate the degree of proximity of reachable sets for small lengths of time intervals. These sets contract to a single-point set as the interval length tends to zero, so the Hausdorff metric is not enough for this purpose. Here we use the concept of asymptotic equality of sets introduced in [4] and based on the Banach-Mazur metric.
The paper is arranged as follows. In Section 1, we introduce the concept of asymptotic equality of sets using the Banach-Mazur metric. We prove several auxiliary statements concerning the connection of this concept with the properties of support functions. In Section 2, we consider relations between the images of a Hilbert ball under nonlinear mapping depending on a small parameter and under its linear approximation. Further, we apply these results to the study of the asymptotic behavior of the reachable sets of nonlinear systems with integral control constraints. We formulate sufficient conditions for the asymptotic equality of reachable sets of nonlinear and linearized systems. These conditions depend on the asymptotic behavior of the controllability Gramian of the linearized system. The asymptotic behavior of the smallest eigenvalue of the controllability Gramian for a time-invariant linear control system with a single input is studied in Section 3. In Section 4, we apply the obtained asymptotics to the study of reachable sets for affine-control nonlinear systems on a small time interval. We give two examples of nonlinear two-dimensional systems and present the results of numerical simulations.

Asymptotic equality of sets
Let X, Y ⊂ R n be convex compact sets. We assume that the zero vector is an interior point of each of these sets. The Banach-Mazur distance ρ(X, Y ) between X and Y is defined by the equality ρ(X, Y ) := log r(X, Y ) · r(Y, X) , r(X, Y ) = inf t ≥ 1 : tX ⊃ Y .
For convex closed sets X and Y , the inclusion tX ⊃ Y holds if and only if where δ(y|X) is the support function of the set X: Hence, we have the formula Note that, due to the condition 0 ∈ int Y , the inequality δ(y|Y ) > 0 holds for y = 0. Suppose further that the sets under consideration depend on a small positive parameter ε, X = X(ε) and Y = Y (ε) are convex compact sets, and the zero vector is an interior point of each of these sets for 0 < ε ≤ ε 0 . We also assume that the multivalued mappings X(ε) and Y (ε) are bounded. The sets X(ε) and Y (ε) are called asymptotically equal [4] if ρ(X(ε), Y (ε)) → 0 as ε → 0.
Formula (2.1) implies the following statement.
To prove the necessity of condition (2.2), suppose, on the contrary, that this condition is violated. Then there exist 1 > σ > 0 and a sequence ε k → 0 such that the following relations are valid for an infinite number of the sequence terms: In the former case, we have r(X(ε k ), Y (ε k )) ≥ 1+σ and, therefore, In the latter case, we obtain This implies that for an infinite number of the sequence terms ε k . This contradicts the convergence of ρ(X(ε k ), Y (ε k )) to zero.
From these inequalities, we obtain relations (2.2) and hence, by Note that the definition of ρ(X, Y ) is symmetrical with respect to the sets X and Y . Therefore, in the statement of the theorem, δ min (Y (ε)) can be replaced by δ min (X(ε)).

Auxiliary results
Let X and Y be Banach spaces. Denote by B X (a, µ 0 ) ⊂ X the ball of radius µ 0 centered at a. Consider a mapping F ε : B X (a, µ 0 ) → Y depending on a parameter ε, 0 < ε < ε 0 . Assumption 1. The mapping F ε (x) has a Fréchet derivative with respect to x, which satisfies the Lipschitz condition on B X (a, µ 0 ) Let a function µ(ε) map (0, ε 0 ] to (0, µ 0 ]. Assume that µ(ε) → 0 as ε → 0. Denote by G ε the image of the ball B X (a, µ(ε)) under the mapping F ε : Theorem 2. Suppose that condition (3.1) holds. Then where h is the Hausdorff distance between sets and co G denotes the convex hull of the set G.
P r o o f. The proof follows from the proof of Theorem 1 in [10].
Let X and Y be real Hilbert spaces. Suppose that a mapping F : X ⊃ B X (a, µ 0 ) → Y is differentiable and its Frechét derivative F ′ satisfies the Lipschitz condition with constant L. Let a mapping F be regular at the point a, i.e., let the operator F ′ (a) : X → Y be a surjection. The latter property implies the existence of a positive number γ such that F ′ (a) * y ≥ γ y for all y ∈ Y , which is equivalent to the inequality Here (·, ·) is the bilinear form for the duality between Y and the space Y * conjugate to Y , F ′ (a) * stands for the operator adjoint to a bounded linear operator F ′ (a). In [17,Theorem 1], it is shown that, if the inequality holds, then the image of the ball B X (a, µ), i.e., the set In what follows, we assume that X is a Hilbert space and Y = R n is a finite-dimensional Euclidean space. Consider the family of operators F ε assuming that each mapping F ε is regular at the point a. Denote by ν(ε) the smallest eigenvalue of the operator (matrix) Note that in this case the set , and ν(ε) is the length of its smallest semiaxis. Theorem 2 implies the following statement.
Thereby, the set G ε − F ε (a) is asymptotically equal to µ(ε)E ε . This means that the image of the ball B(a, µ(ε)) under the nonlinear transformation F ε is close in shape to the ellipsoid F ε (a) + µ(ε)E ε . The latter is the result of transforming the ball by means of a linear approximation of F ε at the point a.

Small-time reachable sets
Consider a nonlinear control-affine systeṁ where x ∈ R n and u ∈ R r are state and control inputs, respectively, andε > 0. The initial state x 0 is assumed to be fixed. Denote by L 2 [0,ε] the Hilbert space of square integrable functions [0,ε] → R r . Constraints on controls are given in the form where B(0, µ) := u(·) ∈ L 2 [0,ε] : (u(·), u(·)) ≤ µ 2 is a ball of radius µ > 0 centered at zero and (u(·), u(·)) = u(·)) 2 Suppose that, for any u(·) ∈ B(0, µ), there exists a unique solution x(t, u(·)) of system (3.3), this solution is defined on [0,ε], and all trajectories starting from x 0 and corresponding to the controls from the ball B(0, µ) belong to a compact set D. Assume also that the functions f 1 and f 2 have Lipschitz continuous derivatives on D.
Since u(·) L 2 [0,ε] ≤ u(·) L 2 [0,ε] , the set G(ε, µ) can be written as follows: We study the behavior of reachable sets G(ε, µ) under the assumption that ε is a small number. Using a time change, we reduce the problem of describing reachable sets on the time interval [0, ε] to a similar problem on the interval [0, 1] for another system whose equations and integral constraints on the control depend on ε.
Differentiating equality (3.8), it is easy to see that W ε (t) is a solution of the linear differential equationẆ The system is completely controllable on the interval [0, 1] if and only if W ε (1) is positive definite. It is known (see, for example, [14,16]) that, in this case, the reachable set under the constraint is an ellipsoid defined as the set of solutions of the inequality x ⊤ W −1 ε (1)x ≤ µ 2 . From the above, we can conclude that the matrix W ε = F ′ ε (0)F ′ ε (0) * coincides with the controllability Gramian W ε (1) of system (3.6) and the ellipsoid µ(ε)E ε =Ĝ(ε, µ) is the reachable set at time 1 of system (3.6) under constraint (3.5). Note that F ε (0) equals to x(ε, 0). Taking into account that µ(ε) = µ √ ε, we arrive at the following statement.
Using the reverse time change, it is easy to show thatĜ(ε, µ)) is the reachable set at time ε for the linearized system (3.3). Thus, Theorem 3 states that, under proper asymptotic behavior of the smallest eigenvalue of the controllability Gramian, the small-time reachable set is asymptotically equal to the reachable set of the linearized system. The asymptotic behavior of the Gramian for the case of linear autonomous systems is studied in the next section.

Asymptotics of the smallest eigenvalue of the controllability Gramian
Consider a linear time-invariant control systeṁ where x ∈ R n , u ∈ R r , and ε > 0 is a small parameter. If the pair (A, B) is completely controllable, then (εA, B) is also controllable for all ε = 0. In this case, the smallest eigenvalue of the controllability Gramian ν(ε) = ν(W ε (1)) is positive for all ε > 0. In this section, we study the asymptotic behavior of ν(ε) for small ε.
Consider the controllability Gramian W ε (t) of system (4.1). The matrix W ε (t), t > 0, is positive definite for every ε = 0 if and only if the pair (A, B) is completely controllable. Let us look for W ε (t) as the sum of series in powers of ε: Differentiating (4.2) and equating coefficients at equal powers of ε, we geṫ Integrating equations (4.3), we get In view of the estimate U k ≤ 2 A U k−1 ≤ 2 k A k U 0 , series (4.4) and (4.2) are majorized by the converging series Here A is the spectral matrix norm induced by the Euclidean vector norm. As a result, we find that the matrix W ε = W ε (1) is represented as the sum of series (4.4) uniformly convergent on every bounded subset of R.
for all ε.
Consider systems with single control. In this case, A is an n×n matrix and B is an n-dimensional column-vector.
Theorem 4. [9, Theorem 1] Assume that a system is completely controllable. If n = 2, then there exist α > 0 and β > 0 such that the following inequality holds for all sufficiently small ε > 0: If n ≥ 3, then there exists β > 0 such that for all sufficiently small ε > 0.
The proof of this theorem is based on reducing the control system to the Frobenius form.

Small-time reachable sets of time-invariant systems
Consider an autonomous control system with a single inpuṫ where x ∈ R n , u ∈ R, f : R n → R n is a continuously differential mapping, B is an n × 1 matrix (a column-vector), and x 0 is a fixed initial state, with control variables subjected to the quadratic integral constraints Suppose, as above, that there exists a compact set D ⊂ R n containing all trajectories of system (4.5) and that f (x) has a Lipschitz continuous derivative on this set.
Note than the sufficient conditions for the convexity of G(ε) are not satisfied for a system with a single input for n ≥ 3.

Examples
As an illustrative example, consider the Duffing oscillatoṙ which describes the motion of a nonlinear stiff spring on impact of an external force u, with integral constraints ε 0 u 2 (t)dt ≤ µ 2 and zero initial state x 1 (0) = 0, x 2 (0) = 0. Consider the Lyapunov-type function Differentiating V (x(t)) along an arbitrary trajectory of the system and applying an analog of Grownwall's Lemma [23], we find that all trajectories of system (4.6) belong to the compact set D = {x ∈ R 2 : V (x) ≤ µ 2 ε} (see [22]). is completely controllable. From Corollary 3 it follows that, for small ε, the reachable sets G(ε) in this example are convex sets close in shape to ellipsoids.
The results of the numerical simulation are shown in the figure that follows. These results are obtained with the use of an algorithm based on Pontryagin's maximum principle for boundary trajectories. Fig. 1 shows the results of numerical simulation for this example. Its left-hand side exhibits the plot of the boundaries of the reachable set at times ε = 0.5, 0.7, 0.9, 1.2, and 1.5, respectively. A larger set in the figure corresponds to a larger value of ε. This plot indicates that the reachable sets for smaller values of ε are convex and look like ellipsoids. The right-hand side of the figure corresponds to smaller ε. Here the boundaries of reachable sets of the nonlinear system are shown in blue and of the linearized system in red. Note that the reachable sets contract to zero as ε → 0. In order to make the picture more informative, we multiply each of the sets by a scaling factor s(ε) depending on ε. The resulting ellipsoids tend to a degenerate ellipsoid (vertical segment) as ε → 0.
All trajectories of the system belong to a compact set on the plane. This fact could be easily proved by using the transition to the polar coordinates. The matrices A and B of the system linearized along the trajectory x(t) ≡ (1, 0) have the following form: