External Ellipsoidal Approximations for Set Evolution Equations

In many applications, uncertainty and imprecision in control systems require the focus on reachable sets instead of single state vectors. Then, closed-loop controls also refer to these attainable sets leading to a class of set evolution problems. We suggest sufficient conditions for its well-posedness and for approximating their solutions by intersections of finitely many time-dependent ellipsoids characterized by solutions to a system of ordinary differential equations.


Introduction
The focus is the external ellipsoidal approximation of all attainable states in a control system with uncertainties as they occur in many technical applications. This arouses our interest in reachable sets (instead of single state vectors) and we face the challenge that these sets have a feedback influence on the underlying control system and thus on their own evolution. The results here lay the foundations for future descent methods in optimization problems.
Communicated by Felix L. Chernousko. As a first motivation, we sketch the optimization problem of output feedback control (OFC) in uncertain systems introduced by Kurzhanski and Varaiya [35] (see also [36,Ch. 10]). In its linear (or linearized) form without available measurements, the state x(t) ∈ R n is determined by x = A(t) x +B(t) η+C(t) v with a control v(t) ∈ V and bounded "unknown noise" η(t). Furthermore, the initial set K 0 ⊂ R n and the setvalued map U (·) of constraints on η(t) are given. For each control v ∈ L 1 ([0, T ], V ), the guaranteed state estimation K v (t) ⊂ R n at time t ∈ [0, T ] consists of the values x(t):=x(t; 0, x 0 ) of all solutions x : [0, t] −→ R n to x = A x + B η + C v with x(0) = x 0 ∈ K (0) and any measurable noise η(t) ∈ U (t) (e.g., [36,Ch. 9]). The key goal of OFC is to specify a control strategy v = v(t,K ) ∈ V in terms of the time t ∈ [0, T ] and a set-valued state estimationK ⊂ R n which "for any starting position (t 0 , K 0 ), 0 ≤ t 0 < T , would bring x(T ; t 0 , x 0 ) ∈ R n [of a solution x with x(t 0 ) = x 0 ] to a preassigned neighborhood of the given target set M ⊂ R n at given time T -whatever" the "uncertain item" η(s) ∈ U (s) (s ∈ [t 0 , t]) is [36, p. 375]. In terms of set-valued analysis, K v (t) is the reachable set of the initial set K 0 ⊂ R n and the differential inclusion x ∈ A(s) x + B(s) U (s) + C(s) v(s) at time t and, the OFC problem focuses a closed-loop control v = v(t,K ) depending on a setK of states (and not a state vector). A relaxed OFC problem can be formulated as an optimal control problem: Minimize = sup ξ ∈ K (T ) dist(ξ, M) ≥ 0 over all set-valued maps K : [0, T ] R n and measurable controls v ∈ V such that each K (t) coincides with the reachable set of K 0 and x ∈ A(s) x + B(s) U (s) + C(s) v at time t. Indeed, if the minimum is attained and smaller than a given threshold ρ > 0 then K (T ) is contained in the ρ-neighborhood of M (as demanded originally). This problem is already solved by Kurzhanski and Varaiya [35]-even with some generalizations. We mention it here because we consider it an excellent example of how useful sets (as states) instead of vectors can be for handling bounded (but) unknown perturbations deterministically. This is the first step to a broad class of optimization problems. Further examples of set-oriented descent methods with applications in image processing are discussed in, e.g., [22,39,40,66]. In general, reachable sets play an important role in deterministic systems with (bounded) "unknown noise" or other forms of lacking information (about initial states and parameters) because they are the smallest set containing all attainable states whatever the bounded uncertainties are (see, e.g., [10]). For nonlinear differential inclusions, several established numerical methods are related to grids (e.g., [6,8,28,57]) or based on the level-set formulation and its HJB equation (e.g., [38,45,46,65]). Hence, they are usually expensive and suffer from the "curse of dimensionality". For linear differential inclusions, the situation is different as the convexity of initial sets is obviously preserved. Chernousko, Kurzhanski and others suggest ellipsoids for approximating these convex sets as supersets and subsets, respectively (see, e.g., [13][14][15][16]23,31,33,34,36]). This special subclass of convex sets has the advantage that the evolution of an ellipsoid can be described in form of ordinary differential equations (ODEs) for its center and the underlying positive definite matrix. Hence, it reduces the numerical effort in high dimensions significantly. As the price to pay, however, this approach is usually applied to linear (or linearized) differential inclusions. For approx-imating solutions within any given collection of sets, Quincampoix and Veliov suggest a general framework in [55] (but without a concrete numerical algorithm).
This article aims at a rather cheap numerical approximation algorithm for a larger class of set evolution problems: In comparison with Kurzhanski's OFC problem, the closed-loop control v = v(t,K ) has nonlinear influence on the state vector x(t), i.e., the linear control equation x = A(t) x + B(t) η + C(t) v (with bounded "unknown noise" η(t) ∈ U (t)) is replaced by x = A(t, v) x + B(t, v) η. Then, the same ansatz v = v(t,K ) leads to a differential inclusion x (t) ∈ A t, K (t) x(t)+B t, K (t) U (t) whose coefficient matrices depend on its own reachable set K (t). We extend the ellipsoidal approximation which Kurzhanski et al. originally developed for linear differential inclusions. In particular, our sufficient conditions on the coefficients concern two aspects: Firstly, this set evolution problem is well posed. Secondly, the inclusion property is preserved, i.e., whenever K (0) is contained in an ellipsoid E(0), then each of them evolves independently of the other in such a way that K (t) ⊂ E(t) holds for every t. In the following, the well-posedness of the set evolution problem and the inclusion property of any two solutions are handled even for nonlinear differential inclusions and compact (not necessarily convex) sets. This part extends various results in, e.g., [3,4,[17][18][19]37,41,44,49,53,54,62]. Then, in favor of fast numerical methods, the external approximation by ellipsoids is restricted to differential inclusions x ∈ A(·, K ) x + B(·, K ) U (i.e., linear in x and η ∈ U ). It is worth mentioning that its linear aspects do not concern the compact set K (t) ⊂ R n or its set properties. We present sufficient conditions on the matrix coefficients such that the wanted set K (t) is contained in the intersection of finitely many ellipsoids whose time-dependent centers and positive definite matrices solve an ODE system. The Pompeiu-Hausdorff distance between K (t) and the intersection can be made arbitrarily small by choosing the numbers of ellipsoids sufficiently large. We also give an example that K (t) might not have joint boundary points with each of the ellipsoids (as known in the classical case of linear inclusions, see Proposition 3.1 (2.) below). This article is structured as follows. First, we summarize the notation used below. Section 2 specifies how the evolution of compact sets in time can be characterized in various (but equivalent) ways. It lays the foundations of what we call set evolution equations and, we give results about their initial value problems (IVP). In Sect. 3, we summarize the method by Kurzhanski et al. for external ellipsoidal approximations of solutions to linear control systems. Then, it is extended to a new class of set evolution problems and, a nonlinear ODE system is suggested for any given number of ellipsoids whose intersections serve as approximations. Section 4 contains a numerical example. All proofs are collected in Sect. 5.
Notation For any dimension, n ∈ N, K(R n ) denotes the set of all nonempty compact subsets of R n . K co (R n ) abbreviates the set of all nonempty compact convex subsets of R n . · is the Euclidean norm on R n , · op the related matrix norm. On the basis of the so-called Pompeiu-Hausdorff excess both K(R n ) and its subset K co (R n ) are usually supplied with the Pompeiu-Hausdorff metric These metric spaces are known to be complete, locally compact and thus separable (e.g., [4,7,30,58]). The gap between A, Def.
= inf x ∈ A inf y ∈ B x − y . Moreover, we use the same arrow for a set-valued (or multivalued) map as, e.g., [4,5], i.e., for any nonempty sets Y , Z given, g : Y Z is a mapping relating each element y ∈ Y to a subset g(y) ⊂ Z , which might consist of more than one element of Z . A set-valued map is called a tube whenever it is defined on a subinterval of R and has nonempty set values. Further properties of its set values like compactness are usually mentioned explicitly (if required). L n denotes the Lebesgue measure on R n . Set B R := x ∈ R n x < R and B R := x ∈ R n x ≤ R for R ≥ 0. Solutions to ordinary differential equations or inclusions are usually understood in the sense of Carathéodory (unless stated otherwise).

Reachable Sets Plus Feedback Lead to Set Evolution Equations
Consider a control system x ∈ g(t, x, U ) (a.e.) where a function g : [0, T ] × R n × R m −→ R n and a nonempty control subset U ⊂ R m are given. The reachable set of an initial set K 0 ⊂ R n at time t ∈ [0, T ] is defined as Under appropriate assumptions about g and U , each absolutely continuous solution . Hence, we prefer the focus on differential inclusions instead of ordinary differential equations with time-dependent control. From the conceptual point of view, reachable sets can be interpreted as a way of "integrating" nonempty (usually closed) subsets w.r.t. time. In the special case, for example, that g does not depend on the state vector x ∈ R n explicitly, i.e., whenever g = g(t, u), the reachable set R g (t, K 0 ) can be expressed in terms of an Aumann integral (w.r.t. L 1 ) R g(·,U ) (t, K 0 ) = K 0 + relationship between set integrals and reachable sets in terms of generalizations are in, e.g., [40,42,43]). In the next step, we aim at an additional feedback w.r.t. the current compact subset of R n . This extension is motivated by Kurzhanski's OFC problem (mentioned in the introduction) and by examples of descent methods in image segmentation, nonlocal agents-population interaction with closed-loop strategies and deterministic approaches to robust control problems (see, e.g., [17,22,40,43,66]). Assume that the right-hand side of the differential inclusion depends on a further argument, namely a nonempty compact subset of the state space R n , i.e., we consider the function f : In particular, the feedback mentioned previously concerns the differential inclusions which depends on K (·). Sufficient conditions of well-posedness and several examples are investigated in, e.g., [17].

Differential Characterizations of These Compact-Valued Solutions
Reachable sets of differential inclusions are also characterized by means of the so-called integral funnel equation. This approach is very popular among Russian mathematicians like Filippova, Kurzhanski, Panasyuk, Tolstonogov and collaborators (e.g., [34,48,[50][51][52]61,62] and related references). Indeed, consider the compact initial set K 0 ⊂ R n , the nonempty control set U ⊂ R m and the function g : [0, T ]×R n ×U −→ R n given. Under appropriate assumptions, the compact-valued tube of reachable sets K : at a.e. time instant t ∈ [0, T ). Furthermore, slightly stronger hypotheses about U , g even guarantee that it is the only Lipschitz continuous compact-valued tube K : Aubin suggests an alternative criterion of differential type and chooses it as the starting point of his so-called morphological equations (in the metric space (K(R n ), d)) (see, e.g., [2][3][4]). At time instant t ∈ [0, T ) and for a short period h > 0, it is now the reachable set R g(t,·,U ) h, K (t) ⊂ R n of the autonomous differential inclusion y (s) ∈ g t, y(s), U ) a.e. in [0, h] which induces an approximation of K (t +h) ⊂ R n . Similarly to time derivatives of curves in a normed vector space, the distance between them is to vanish "in first order" for h ↓ 0.
In shape sensitivity analysis and shape optimization, the special case of U ⊂ R m consisting of just a single vector leads to the so-called shape derivatives used by Delfour, Sokoowski, Zolésio and others in the so-called velocity method (see, e.g., [20,21,59] and references therein). In more detail, sufficient conditions on U ⊂ R m and g : Hence, we have three criteria on the reachable sets of a nonautonomous differential inclusion x ∈ g(·, x, U ) and a compact initial set K 0 ⊂ R n . Now we implement the additional aspect of set feedback (again) and obtain the following result about set evolution equations. Shortly speaking, it represents a special case of [42,Theorem 1] which concerns closed-valued tubes evolving along nonautonomous evolution inclusions in a separable Banach space (instead of R n ) and thus, we do not give a proof in detail. Def.
= f (t, x, M, u) u ∈ U ⊂ R n is closed and convex. (ii) (measurable in t) For all x ∈ R n , M ∈ K(R n ) and u ∈ U , f ( · , x, M, u) :

Remark 2.4
From now on, we do not really distinguish between two established concepts, i.e., quasidifferential equations by Panasyuk applied to K(R n ), d (e.g., [49,53,54]) and morphological equations by Aubin which are the example of his mutational equations applied to K(R n ), d and the "transitions" induced by reachable sets (see, e.g., [3,4,41]). Indeed, Proposition 2.1 states their equivalence under suitable assumptions. Several publications characterize the solution to a set evolution equation in terms of the Hukuhara derivative (w.r.t. time), and thus, the tubes are always assumed to be convex-valued (see, e.g., [37,44,62] and related references). Under suitable assumptions about f : R n "solves" the differential equation D H K (t) = f t, K (t), U (in that sense) if and only if it fulfills the following condition on Aumann integrals for every t ∈ [0, T ] As a consequence, we consider that concept as the special case of our approach in which the function f : [0, T ] × R n × K(R n ) × U −→ R n does not depend on the state x ∈ R n explicitly. Indeed, the right-hand side in Eq. (4) coincides with the reachable set of K (0) ⊂ R n and y ∈ f · , K (·), U at time t (as mentioned in Sect. 2.1). It is worth mentioning that even in the autonomous linear case, state x ∈ R n and set K (s) ⊂ R n cannot be simply exchanged with each other. Indeed, Tolstonogov gives the example in R [62, p. 209 f.] that the reachable interval of the single initial state 0 ∈ R and x ∈ −αx + U (with α > 0 and U = −U ⊂ R having more than one element) does not coincide with the solution R n satisfy the following a priori bound with R:

Ellipsoidal Approximations of Reachable Sets for Linear Control Systems
For several decades, ellipsoids have been very popular for approximating convex compact subsets of R n . Russian mathematicians, in particular, like Chernousko, Filippova, Kurzhanski and collaborators have proposed them for estimating reachable sets of control systems that are usually linear or linearized (see, e.g., [13][14][15][16]23,31,33,36] and related references).
Their key advantage is the simple algebraic characterization: Each (so-called nondegenerate) ellipsoid is determined completely by its center p ∈ R n and its matrix Q ∈ R n×n (symmetric and positive definite) We start with a linear time-variant control system x ∈ A(·) x + B(·) U (a.e. in [0, T ]) with A : [0, T ] −→ R n×n and B : [0, T ] −→ R n×m given (as in, e.g., [33,36]). For the sake of simplicity, the ellipsoidal control set U :=E(q u , Q u ) ⊂ R m does not depend on time. For each initial K 0 ∈ K(R n ), the tube K : [0, T ] R n of reachable sets is characterized by the integral funnel equation In general, the reachable set K (t) ⊂ R n is not an ellipsoid though-even if K 0 is one. . Its minimal property (2.) (c) indicates in which sense these ellipsoids are "optimal approximations" of R A · +B U (t, K 0 ) within their class.
Suppose that the initial set K 0 :=E(x 0 , X 0 ) ∈ K(R n ) and the control set U :=E(q u , Q u ) ⊂ R m are non-degenerate. Then, the following statements hold:

Some External Approximation for a Solution to a Set Evolution Equation
Now we aim to extend these results from linear time-variant control systems (and their reachable sets) to a class of set evolution equations (and their solution tubes in the sense of Definition 2.3). Hence, the right-hand side of the set evolution equation (described by f in Sect. 2.2) is now supposed to be linear w.r.t. x and u. Reachable sets of nonautonomous linear differential inclusions are known to be always convex as a consequence of the variations of constants formula. Thus, we focus on convex compact subsets of R n instead of K(R n ). In analogy to the notation in Sect. 3.1, let the coefficient functions be given. We consider the set evolution equation with the function and U ⊂ R m satisfy the following conditions: .
Then, the unique solution tube K : [0, T ] R n of the IVP

A Computational Method for an External Approximation With Ellipsoidal Values
Proposition 3.1 (2.) provides an ODE system which specifies an ellipsoid-valued tube as an external approximation of the reachable set. In more detail, it concerns the reachable set R A · +B U (t, K 0 ) of a nonautonomous linear differential inclusion x ∈ A(t) x + B(t) U and, the ODE system describes the evolution of the center x(t) ∈ R n and the positive definite symmetric matrix X (t) ∈ R n×n of the time-dependent ellipsoids. Set evolution equations are essentially based on the notion that the coefficients depend on the current set in addition: respectively. This gist motivates us to consider the following nonlinear ODE system . Strictly speaking, Proposition 3.2 considers the intersection of finitely many ellipsoids as an external approximation of the solution value K (t) ⊂ R n . Assumption 3.2 (vii) indicates how to choose the coefficients appropriately, i.e., in terms of their pointwise intersection E ∩ (t). It leads directly to ODE system (7) below which is easy to solve numerically (using standard methods for the support function of E ∩ (t)).

Proposition 3.3 Let
For j = 1, . . . , N , let 0 j ∈ R n \{0}, x 0 j ∈ R n and positive definite symmetric X 0 j ∈ R n×n be given such that N k=1 E(x 0k , X 0k ) ⊂ R n has nonempty interior. Consider the ODE system with the abbreviations Then, the following statements hold: N ). Moreover, for all t ∈ [0, T ] : In regard to external approximations for set evolutions, Proposition 3.2 has the direct consequence:

No Minimal Property of This Ellipsoidal Approximation in General
In the established context of linear differential inclusions, Proposition 3.1 (2.) provides the connection between solutions to an ODE system for x(·), X (·) and the ellipsoid- In connection with the more general problem class of set evolution equations, however, we have not made any comment on the last two features so far, i.e., The following example shows that such a form of minimality does not hold under the assumptions of Proposition 3.3. In a word, the current set K (t) ⊂ R n might have a significant influence on the coefficient matrices A t, K (t) , B t, K (t) ∈ R n×n such that joint boundary points are lost instantaneously.  1) and (−1, 1) are eigenvectors of X 0 associated with the eigenvalues 2, 4, respectively.) The variation of constants formula provides an explicit representation of the reachable set R(t) ⊂ R 2 of the autonomous linear differential inclusion In particular, R(t) ⊂ R 2 is convex, compact, but not an ellipsoid for t > 0.
is the solution to the IVP (6). Fixing a unit vector 0 ∈ R 2 arbitrarily, Proposition 3.3 and Corollary 3.4 provide an ellipsoid-valued tube E : with the following properties: with the initial values (0) = 0 and X (0) = X 0 Def.
(The equation of the center x(·) in ODE system (7) has the unique solution x(·) = 0 in this example and so, we do not mention it explicitly any longer.) Indeed, consider the solution X : (The adjoint equation for is the same as before: Hence, it remains to verify that E(t) is contained in the interior E(t) • for every t ∈ (0, 1]. As the reachable set R(t) is not an ellipsoid, we have R(t) E(t) and so e E(t), and thus, d ds Hence, the same steps as before lead to

As a consequence, E(t)
Def.
In particular, the reachable tube R(·) is the solution of the underlying IVP (6), but R(t) cannot have a joint boundary point with E(t) for any t ∈ (0, 1].

Approximating the Solution of a Set Evolution Equation with Arbitrary Precision
Example 3.5 shows that the value K (t) of the solution tube might be contained in the interior of any approximating ellipsoid E x(t), X (t) based on the ODE system (7), (8). Whenever a joint boundary point (as mentioned in Proposition 3.1 (2.) (d)) does not exist, it is not so obvious how to estimate the gap of the approximation.
The following results states that the exact solution can be approximated with arbitrary precision-by choosing the number of ellipsoids sufficiently large.

A Numerical Example
This example is deliberately short and in two dimensions so that numerical results can be shown in figures. Motivated by challenges of collision avoidance, we consider a simple cart under the influence of bounded "unknown noise" and aim at a guaranteed state estimation of its position and velocity. In addition, security reasons require a safety zone which grows with the "uncertainty" of the state estimation. Now we suggest a simple model for this situation. Initially (i.e., without any noise or safety zone), the cart moves according to the linear control system for scalar position

Tools about Reachable Sets of Differential Inclusions
) For every initial set M 0 ∈ K(R n ), the Pompeiu-Hausdorff distance between the reachable sets of the autonomous inclusion y ∈ g(t, y, U ) and the nonautonomous inclusion y ∈ g(t + ·, y, U ) satisfies lim R n is Lipschitz continuous w.r.t. d and, its Lipschitz constant is ≤ 1 + r + T · e T .
As a consequence of well-known Filippov's theorem about solutions to differential inclusions, the following bound holds for the Pompeiu-Hausdorff distance between reachable sets (see, e.g., the proofs of [ . For all initial sets K 1 , K 2 ∈ K(R n ) with K 1 ∪ K 2 ⊂ B r , the following estimate with R:= r + T · e T holds at each time t ∈ [0, T ]

Proof of Proposition 2.5 Consider
for a.e. t ∈ [0, T ] and, Gronwall's inequality leads to the claimed estimate.

Inclusion Principle of Solution Tubes (Proposition 2.7)
The gist of the proof is to reformulate the condition K ⊂ M 1 ∩ M 2 as a constraint on tuples (K , Weak invariance (a.k.a. viability) has already been investigated by Aubin and Gorre (e.g., [4, § 4.3.3], [26,27]). Now we use some of their technical results for verifying the (strong) invariance of C.

Lemma 5.5 (Gorre [4, Theorem 4.2.8] [26])
Let U ⊂ R m be nonempty compact and g 1 , g 2 , g 3 : R n × U −→ R n satisfy the following conditions: (i) For all x ∈ R n , the set g j (x, U ) ⊂ R n is compact and convex.
(ii) For every x ∈ R n , g j (x, · ) : U −→ R n is continuous.
(iii) There exists λ > 0 such that for each u ∈ U , g j ( · , u) : respectively, such that for every ∈ N, The next lemma extends (forward) Lebesgue points to measurable functions with values in a metric space Y .

Proof of 2.7
We adapt the arguments usually used for (strong) invariance theorems of differential equations or inclusions (see, e.g., [  = f (t, x, S, u) u ∈ U is compact due to continuity assumption 2.1 (iii) and the compactness of U .
In particular, for every • δ(·) is differentiable at t • the inclusion condition 2.7 (vii') holds at t and and the same for g, g.

Proof of Proposition 3.2 Set
We consider the auxiliary function and aim at verifying δ(t) = 0 for all t ∈ [0, T ]. δ(0) = 0 holds due to the assumption At every time instant t ∈ [0, T ], the convex set K (t) ⊂ R n coincides with the reachable set of K 0 and the nonautonomous linear differential inclusion x ∈ A s, K (s) x + B s, K (s) U (a.e.), which can be represented by means of the variation of constants formula (see, e.g., [33, § § 1.1, 1.2]). As a consequence, each K (t) ⊂ R n has in common with K (0) = K 0 that its interior is not empty. Furthermore, K (·) is Lipschitz continuous. Hence, there exists a radius r 0 > 0 such that for every t ∈ [0, T ], K (t) contains a closed ball with radius 2 r 0 . Choose > 0 sufficiently small such that its product with the maximum of the Lipschitz constants of K (·), From now on, we focus on the restriction of δ to Then, there is a closed subset J ε of [t 0 , T ) with the following properties: j = 1, . . . , N ) satisfies condition 3.2 (vii). • Each τ ∈ J ε is a (forward) Lebesgue point of the characteristic function χ J ε : Choose any τ ∈ J ε . Then, the excess condition 3.2 (vii) also holds for E ∩ (τ ), i.e., Indeed, the autonomous differential inclusion x ∈ A τ, H (τ ) x + B τ, H (τ ) U induces the auxiliary tube R : [0, ∞) R n of reachable sets of H (τ ) ⊂ R n . R(·) satisfies the integral funnel condition at (even) every time h ∈ [0, ∞) as a consequence of Proposition 2.1. Furthermore, Proposition 2.5 and Assumptions 3.2 (iii), (iv) lead to this upper bound for each h ∈ [0, The characterization of J ε and the continuity of H (·) imply lim i.e., H satisfies the integral funnel condition at time τ .
In the next step, we conclude from the triangle inequality for every h ∈ 0, T − t As ε ∈ (0, T − t 0 ) had been chosen arbitrarily, the last inequality holds for a.e. τ ∈ [t 0 , T ] and so, Gronwall's inequality implies δ = 0 in [t 0 , T ], i.e., Finally, it is worth mentioning that T was chosen as T = min t 0 + , T with > 0 depending only on r 0 and the Lipschitz constants of K (·), E 1 (·), . . . , E N (·) (but not on t 0 ). As a consequence, we conclude δ = 0 in the whole interval [0, T ] by means of finitely many subintervals of this form [t 0 , T ] (chosen in a piecewise way).

A Computational Method for an External Approximation With Ellipsoidal Values (Proposition 3.3)
On our way to proving Proposition 3.3, we consider the required property N k=1 E x k (t), X k (t) = ∅ a technical challenge since its relationship with the ODE systems (for all x j (·), X j (·)) is not really obvious. Hence, for ρ ≥ 0 fixed arbitrarily, we focus on the following auxiliary problem ( j = 1, , . . . , N ) with the modified abbreviations In comparison with the original problem (7), (8), the intersection E ∩ (t) Def.
and, A is bounded by due to hypothesis 3.3 (iv'). x j (·) is characterized as the solution to an inhomogeneous linear ODE and so, Gronwall's inequality provides an explicit a priori bound. (2.) Q u ∈ R n×n is symmetric by assumption and so is Q B ∩ (t) ∈ R n×n then. Hence, both X j (·) and X j (·) are Carathéodory solutions Y : at a.e. time s ∈ [0, τ ) and hence, d with γ j = const(X 0 j ) > 0, i.e., X j (t) ∈ R n×n is positive definite. (4.) For every t ∈ [0, τ ], the Cauchy-Schwarz inequality and assumption 3.3 (iv') lead to Similarly, we conclude from Eq. (11), Q u ∈ R n×n being positive definite and hypotheses 3.3 (iv'), (vi') In regard to the other scalar product in the quotient π j (t), we have already verified in the proof of the statement (3 is non-decreasing and so, for every t ∈ [0, τ ], Furthermore, its time derivative mentioned there implies for a.e. t ∈ [0, τ ] and thus for all t ∈ [0, τ ] (5.) In combination with Assumptions 3.3 (iv'), (vi'), ODE (10) of X j (·) has the consequence Subsequent Lemma 5.12 proves the aspect of existence in Proposition 3.3 (1.).
Afterward, the statement about uniqueness follows in Lemma 5.14. In preparation for both results, the next lemma indicates a technically essential property of solutions to the "approximating" system (i.e., with ρ > 0) and the "exact" system (i.e., ρ = 0).

Lemma 5.11
In addition to the assumptions of Proposition 3.3, fix ρ ≥ 0 arbitrarily and let j : N ) be solutions to ODE system (10) with initial values 0 j ∈ R n \{0}, x 0 j ∈ R n and positive definite symmetric X 0 j ∈ R n×n given such that N k=1 E(x 0k , X 0k ) ⊂ R n has nonempty interior. Then, there exists a radius r = r ( , T , Then, B r z(t) ⊂ R n is contained in the reachable set R(t) of the differential inclusion x ∈ A ·, E ∩,ρ (·) x + B ·, E ∩,ρ (·) U and the initial set B R 0 (z 0 ) at each time t ∈ [0, T ]. Indeed, for each y t ∈ B r z(t) , the variation of constants formula provides the solution y : [0, t] −→ R n of y = A ·, E ∩,ρ (·) y + B ·, E ∩,ρ (·) q u with y(t) = y t and, Gronwall's inequality implies = N k=1 E x k (t), X k (t) has nonempty interior. • j (t) = 0 and X j (t) ∈ R n×n is symmetric.
• There exist c j , C j > 0 (depending on , Q u ) and c j , C j > 0 (depending on , • X j (·) is Lipschitz continuous with a Lipschitz constant depending only on , Q u , X 0 j , T .
Proof Every solution tuple to an IVP with ODE system (7) also solves the ODE system (10) with ρ = 0 and abbreviations (11). Due to Gronwall's inequality, it is sufficient to prove that the functions on the right-hand side of ODE system (10) are Lipschitz continuous w.r.t. states in a neighborhood of the solution values. In comparison with the uniqueness result in Proposition 2.2, for example, it is worth mentioning that now the states are in the vector spaces • R n for the directions j (t) and the centers x j (t) ( j = 1, . . . , N ) • R n×n for the matrices X j (t) ∈ R n×n of ellipsoids ( j = 1, . . . , N ), but not in the metric space K(R n ) (for the ellipsoids). In contrast to Lemma 5.10, we now assume the existence of solutions in [0, T ] as well as (implicitly) N k=1 E x k (t), X k (t) = ∅ for all t ∈ [0, T ]. Hence, the same arguments (as for Lemma 5.10 (3.),(4.)) lead to constants c j , C j , c j , C j = const( , Q u , X 0 j ) > 0 such for j = 1, . . ., N and all t The smallest eigenvalue of X j (t) ∈ R n×n (t ∈ [0, T ], j ∈ {1, . . . , N }) is bounded from below by c j e −2 T > 0. Furthermore, x j (·) and X j (·) are Lipschitz continuous with a Lipschitz constant depending only on , Q u , x 0 j , X 0 j , T . Hence, there exists an open neighborhood U ⊂ R n×n , · op of the compact set X j (t) t ∈ [0, T ], j ∈ {1, . . . , N } such that each symmetric matrix in U has eigenvalues > min j Lemma 5.11 guarantees some r > 0 such that for all t ∈ [0, T ], a ball with radius r is contained in N j=1 E x j (t), X j (t) . There exists an open neighborhood V ⊂ R n × R n×n N of the compact set x k (t), X k (t) k=1, ...,N t ∈ [0, T ] such that for all j ∈ {1, . . . , N }, p j ∈ R n and symmetric Q j ∈ R n×n with ( p k , Q k ) k=1, ...,N ∈ V, • Q j ∈ U ⊂ R n×n for each j ∈ {1, . . . , N } and • N j=1 E( p j , Q j ) ⊂ R n contains a ball of radius r 2 . Lemma 5.7 (1.) and the characterization of U imply for all ( p k , Q k ) k=1, ...,N , Together with Assumptions 3.2 (ii), (iii) and 3.3 (iv'),(vi'), this inequality leads to all the aspects of (local) Lipschitz continuity w.r.t. state required for concluding the claimed uniqueness of solutions to ODE system (10) from Gronwall's inequality. ) Let E( p 1 , Q 1 ) and E( p 2 , Q 2 ) be ellipsoids in R n . Then, these statements hold:

Approximating the Set Evolution Solution with Arbitrary Precision (Proposition 3.6)
The proof of [12,Theorem] provides several useful supplementary details.
The following lemma results from the observation that every compact convex set in R n is of arbitrary positive reach (in the sense of Federer) (e.g., [ = x ∈ R n dist(x, M) ≤ ε is of class C 1,1 .
Step 2 Constructing the ellipsoid-valued E j (·) and the auxiliary tube R(·).
• E j (t) is minimal in the class of ellipsoids w.r.t. set inclusions, i.e., there is no ellipsoid E ⊂ R n with R(t) ⊂ E E j (t). • j (t) is normal to R(t) and E j (t) in ξ j (t) := x j (t)+ j (t), X j (t) j (t) − 1 2 X j (t) j (t) ∈ ∂ R(t) ∩ ∂ E j (t).
Step 3 An upper bound of the Pompeiu-Hausdorff distance between R(t) and E ∩ (t).

Conclusions
Ellipsoids represent a well-established approach of approximation in control theory and related optimization problems (like the OFC by Kurzhanski and Varaiya in [35], [36,Ch. 10]). Now they are used for external approximations of convex-valued solutions to so-called set evolution equations. In a word, we focus on reachable sets to linear time-variant control systems whose coefficient matrices depend on their own reachable set. (This very general form of feedback can be formulated equivalently in terms of integral funnel equations, Panasyuk's quasidifferential equations and Aubin's morphological equations, see Propositions 2.1 and 2.7). In regard to future applications, the main result consists in sufficient conditions on the coefficients such that the convex solution values can be approximated as intersections of finitely many ellipsoids with arbitrary precision. Their respective centers and matrices are characterized by a nonlinear ODE system (Sect. 3) and so, they can be calculated rather quickly.