On controllability for a nonlinear differentialinclusion

Aurelian CerneaFaculty of Mathematics and Informatics,

University of Bucharest,Academiei 14, 010014 Bucharest, Romania

acernea@fmi.unibuc.ro

Abstract—We consider a nonlinear differential inclusion andwe obtain sufficient conditions for h-local controllability alonga reference trajectory. To derive this result we use convexlinearizations of the nonlinear differential inclusion. More pre-cisely, we show that the nonlinear differential inclusion is h-locally controlable around a solution z if a certain linearizedinclusion is λ-locally controlable around the null solution forevery λ ∈ ∂h(z(T )), where ∂h denotes Clarke’s generalizedJacobian of the locally Lipschitz function h.

I. INTRODUCTION

In this paper we are concerned with the following nonlineardifferential inclusion

x′ ∈ Ax+ F (t, x), x(0) ∈ X0, (1.1)

where X is a separable Banach space, A is a m-dissipativeoperator on X , F : [0, T ] ×X → P(X) is a set valued mapand X0 ⊂ X is a closed set.

Let SF be the set of all C0-solutions of (1.1) and let RF (T )be the reachable set of (1.1). For a C0-solution z(.) ∈ SF andfor a locally Lipschitz function h : X → X we say that thedifferential inclusion (1.1) is h-locally controllable around z(.)if h(z(T )) ∈ int(h(RF (T ))). In particular, if h is the identitymapping the above definitions reduces to the usual concept oflocal controllability of systems around a solution.

The aim of the present paper is to obtain a sufficientcondition for h-local controllability of inclusion (1.1) when Xis finite dimensional. This result is derived using a techniquedeveloped by Tuan for differential inclusions ([7]). Moreexactly, we show that inclusion (1.1) is h-locally controlablearound the mild solution z(.) if a certain linearized inclusionis λ-locally controlable around the null solution for everyλ ∈ ∂h(z(T )), where ∂h(.) denotes Clarke’s generalizedJacobian of the locally Lipschitz function h. The key toolsin the proof of our result is a continuous version of Filippov’stheorem for mild solutions of problem (1.1) obtained in [3] anda certain generalization of the classical open mapping principlein [9].

The paper is organized as follows: in Section 2 we presentsome preliminary results to be used in the sequel and in Section3 we present our main results.

Work supported by the CNCS grant PN-II-ID-PCE-2011-3-0198

II. PRELIMINARIES

Let X be a real Banach space with the norm ‖ · ‖. Denoteby P(X) the family of all nonempty subsets of X and byB(X) the family of all Borel subsets of X . For any subsetA ⊂ X we denote by cl(A) the closure of A.

Let T > 0, I = [0, T ] and denote by L(I) the σ-algebra ofall Lebesgue measurable subsets of I . If A ⊂ I , then χA(·) :I → 0, 1 denotes the characteristic function of A.

As usual, we denote by C(I,X) the Banach space of allcontinuous functions x(·) : I → X endowed with the norm|x(·)|C = supt∈I ‖x(t)‖ and L1(I,X) the Banach space ofall (Bochner) integrable functions x(·) : I → X endowed withthe norm |x(·)|1 =

∫ T0‖x(t)‖dt.

For x, y ∈ X we denote by (x, y)= limh↑0‖x+hy‖2−‖x‖2

2hthe left sided directional derivative of 1

2‖ · ‖2 at x in the

direction y.

Consider A : X → P(X) an operator. With D(A) wedenote its domain. We recall that A is called dissipative if(x1 − x2, y1 − y2) ≤ 0 for any x1, x2 ∈ D(A) and y1 =Ax1, y2 = Ax2.

A is called m-dissipative if it is dissipative and R(I −λA) = X for any (equivalently, for some) λ > 0.

Let A : D(A) ⊂ X → P(X) be m-dissipative and f(·) ∈L1(I,X) which define the differential equation

x′ ∈ Ax+ f(t). (2.1)

A mapping x(·) : I → X is called a strong solution of(2.1) if x(t) ∈ D(A) a.e. on (0, T ), x(·) is locally absolutecontinuous on (0, T ] and there exists g ∈ L1

loc((0, T ], X) suchthat g(t) ∈ Ax(t) a.e. on (0, T ) and x′(t) = g(t) + f(t) a.e.on (0, T ).

It is well-known that if X is reflexive then for any x0 ∈D(A) equation (2.1) has a unique strong solution on I whichsatisfies x(0) = x0 (e.g. [2]). In general, equation (2.1) maynot have strong solutions and a way to overcome this difficultyis the concept of C0-solutions (e.g., [5]).

DEFINITION 2.1. A function x(·) ∈ C(I,X) is called aC0-solution of (2.1) if satisfies: for every c ∈ (0, T ) and ε > 0there exists

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(i) 0 < t1 < ... < c ≤ tn < T , tk − tk−1 ≤ ε, t0 = 0 fork = 1, 2, ..., n;

(ii) f1, f2, ..., fn ∈ X withn∑k=1

∫ tk

tk−1

‖f(t)− fk‖dt ≤ ε;

(iii) v0, v1, ..., vn ∈ X withvk − vk−1

tk − tk−1∈ Avk + fk

for k = 1, 2, ..., n and ‖x(t) − vk‖ ≤ ε for t ∈ [tk−1, tk),k = 1, 2, ..., n.

According to [5], if A is m-dissipative, f ∈ L1(I,X) andx0 ∈ D(A), there exists a unique C0-solution of (2.1) withx(0) = x0.

Denote by x(·, x0, f) : I → D(A) the unique C0-solutionof (2.1) which satisfies x(0, x0, f) = x0.

THEOREM 2.2. ([8]) Let X be a real Banach space, letA : D(A) ⊆ X → X be an m-dissipative operator, let ξ, η ∈D(A) and f, g ∈ L1(I,X). Then for any t ∈ I

‖x(t, ξ, f)−x(t, η, g)‖ ≤ ‖ξ−η‖+∫ t

0

‖f(s)−g(s)‖ds. (2.2)

Next we are concerned with the following Cauchy problem

x′ ∈ Ax+ F (t, x), x(0) = x0, (2.3)

where A is an m-dissipative operator on the separable Banachspace X , F : I ×X → P(X) and x0 ∈ X.

DEFINITION 2.3. A continuous mapping x : I → D(A) issaid to be a C0-solution of problem (2.3) if x(0) = x0 andthere exists f(·) ∈ L1(I,X) with f(t) ∈ F (t, x(t)) a.e. onI and x(·) is a C0-solution on I of the equation (2.1) in thesense of Definition 2.1.

We shall call (x(·), f(·)) a trajectory-selection pair of (2.3)if f(t) ∈ F (t, x(t)) a.e. on I and x(·) is a C0-solution of (2.3).

HYPOTHESIS 2.4. i) F (., .) : I×X → P(X) has nonemptyclosed values and is L(I)⊗ B(X) measurable.

ii) There exists L(.) ∈ L1(I,R+) such that, for any t ∈I, F (t, .) is L(t)-Lipschitz in the sense that

dH(F (t, x1), F (t, x2)) ≤ L(t)||x1 − x2|| ∀x1, x2 ∈ X.

The main tool in characterizing derived cones to reachablesets of semilinear differential inclusions is a certain version ofFilippov’s theorem for differential inclusion (2.3).

Let A : D(A)→ P(X) be a m-dissipative operator.

HYPOTHESIS 2.5. Let S be a separable metric space,X0 ⊂ X is a closed set, a(.) : S → X0 and c(.) : S → (0,∞)are given continuous mappings.

The continuous mappings g(.) : S → L1(I,X), y(.) :S → C(I,X) are given such that for any s ∈ S, y(s)(.) is aC0-solution of

x′ ∈ Ax+ g(s)(t), x(0) ∈ X0

and there exists a continuous function q(.) : S → L1(I,R+)such that

d(g(s)(t), F (t, y(s)(t))) ≤ q(s)(t) a.e. (I), ∀ s ∈ S.

THEOREM 2.6. ([3]) Assume that Hypotheses 2.4 and 2.5are satisfied. Then, there exist M > 0 and the continuousfunction x(.) : S → L1(I,X) such that for any s ∈ S x(s)(.)is a C0-solution of the problem

x′ ∈ Ax+ F (t, x), x(0) = a(s),

satisfying for any (t, s) ∈ I × S

||x(s)(t)− y(s)(t)|| ≤M [c(s)+

||a(s)− y(s)(0)||+∫ t

0q(s)(u)du].

(2.4)

In what follows we assume that X = Rn.

DEFINITION 2.7. A closed convex cone C ⊂ Rn is saidto be regular tangent cone to the set X at x ∈ X ([6]) ifthere exists continuous mappings qλ : C ∩ B → Rn, ∀λ > 0satisfying

limλ→0+

maxv∈C∩B

||qλ(v)||λ

= 0,

x+ λv + qλ(v) ∈ X ∀λ > 0, v ∈ C ∩B.

From the multitude of the intrinsic tangent cones in the lit-erature (e.g. [1]) the contingent, the quasitangent and Clarke’stangent cones, defined, respectively, by

KxX = v ∈ Rn; ∃ sm → 0+, xm ∈ X : xm−xsm

→ v,QxX = v ∈ Rn; ∀ sm → 0+,∃xm ∈ X : xm−x

sm→ v,

CxX = v ∈ Rn; ∀ (xm, sm)→ (x, 0+), xm ∈ X,∃ ym ∈ X : ym−xm

sm→ v

seem to be among the most oftenly used in the study ofdifferent problems involving nonsmooth sets and mappings.We recall that, in contrast with KxX,QxX , the cone CxX isconvex and one has CxX ⊂ QxX ⊂ KxX .

The results in the next section will be expressed, in the casewhen the mapping g(.) : X ⊂ Rn → Rm is locally Lipschitzat x, in terms of the Clarke generalized Jacobian, defined by([4])

∂g(x) = co limi→∞

g′(xi); xi → x, xi ∈ X\Ωg,

where Ωg is the set of points at which g is not differentiable.

Corresponding to each type of tangent cone, say τxX onemay introduce (e.g. [1]) a set-valued directional derivative ofa multifunction G(.) : X ⊂ Rn → P(Rn) (in particular of asingle-valued mapping) at a point (x, y) ∈ graph(G) as follows

τyG(x; v) = w ∈ Rn; (v, w) ∈ τ(x,y)graph(G), v ∈ τxX.

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We recall that a set-valued map, A(.) : Rn → P(Rn)is said to be a convex (respectively, closed convex) processif graph(A(.)) ⊂ Rn × Rn is a convex (respectively, closedconvex) cone. For the basic properties of convex processes werefer to [1], but we shall use here only the above definition.

HYPOTHESIS 2.8. i) Hypothesis 2.4 is satisfied and X0 ⊂Rn is a closed set.

ii) (z(.), f(.)) ∈ C(I,Rn) × L1(I,Rn) is a trajectory-selection pair of (1.1) and a family P (t, .) : Rn → P(Rn),t ∈ I of convex processes satisfying the condition

P (t, u) ⊂ Qf(t)F (t, .)(z(t);u)∀u ∈ dom(P (t, .)), a.e. (I)(2.5)

is assumed to be given and defines the variational inclusion

v′ ∈ Av + P (t, v). (2.6)

Remark 2.9. We note that for any set-valued map F (., .),one may find an infinite number of families of convex processP (t, .), t ∈ I , satisfying condition (2.5); in fact any familyof closed convex subcones of the quasitangent cones, P (t) ⊂Q(z(t),f(t))graph(F (t, .)), defines the family of closed convexprocess

P (t, u) = v ∈ Rn; (u, v) ∈ P (t), u, v ∈ Rn, t ∈ I

that satisfy condition (2.5). One is tempted, of course, totake as an ”intrinsic” family of such closed convex process,for example Clarke’s convex-valued directional derivativesCf(t)F (t, .)(z(t); .).

We recall (e.g. [1]) that, since F (t, .) is assumed to beLipschitz a.e. on I , the quasitangent directional derivative isgiven by

Qf(t)F (t, .)((z(t);u)) = w ∈ R;

limθ→0+

1

θd(f(t) + θw, F (t, z(t) + θu)) = 0. (2.7)

In what follows B or BRn denotes the closed unit ball inRn and 0n denotes the null element in Rn.

Consider h : Rn → Rm an arbitrary given function.

DEFINITION 2.10. Inclusion (2.3) is said to be h-locallycontrollable around z(.) if h(z(T )) ∈ int(h(RF (T ))).

Inclusion (2.3) is said to be locally controllable around thesolution z(.) if z(T ) ∈ int(RF (T )).

Finally a key tool in the proof of our results is the followinggeneralization of the classical open mapping principle due toWarga ([9]).

For k ∈ N we define

Σk := γ = (γ1, ..., γk);k∑i=1

γi ≤ 1, γi ≥ 0, i = 1, k.

LEMMA 2.11. ([9]) Let δ ≤ 1, let g(.) : Rn → Rm be amapping that is C1 in a neighborhood of 0n containing δBRn .Assume that there exists β > 0 such that for every θ ∈ δΣn,βBRm ⊂ g′(θ)Σn. Then, for any continuous mapping ψ :δΣn → Rm that satisfies supθ∈δΣn

||g(θ) − ψ(θ)|| ≤ δβ32 we

have ψ(0n) + δβ16BRm ⊂ ψ(δΣn).

III. THE MAIN RESULT

In what follows C0 is a regular tangent cone to X0 at z(0),denote by SP the set of all C0-solutions of the differentialinclusion

v′ ∈ Av + P (t, v), v(0) ∈ C0

and by RP (T ) = x(T ); x(.) ∈ SP its reachable set at timeT .

THEOREM 3.1. Assume that Hypothesis 2.8 is satisfied andlet h : Rn → Rm be a Lipschitz function with Lipschitzconstant l > 0.

Then inclusion (2.3) is h-local controllable around thesolution z(.) if

0m ∈ int(λRP (T )) ∀λ ∈ ∂h(z(T )). (3.1)

Proof: By (3.1), since λRP (T ) is a convex cone, itfollows that λRP (T ) = Rm ∀λ ∈ ∂h(z(T )). Therefore usingthe compactness of ∂h(z(T )) (e.g. [4]), we have that for everyβ > 0 there exist k ∈ N and uj ∈ RP (T ) j = 1, 2, ..., k suchthat

βBRm ⊂ λ(u(Σk)) ∀λ ∈ ∂h(z(T )), (3.2)

where

u(Σk) = u(γ) :=

k∑j=1

γjuj , γ = (γ1, ..., γk) ∈ Σk.

Using an usual separation theorem we deduce the existenceof β1, ρ1 > 0 such that for all λ ∈ L(Rn,Rm) withd(λ, ∂f(z(T ))) ≤ ρ1 we have

β1BRm ⊂ λ(u(Σk)). (3.3)

Since uj ∈ RP (T ), j = 1, ..., k, there exist (wj(.), gj(.)),j = 1, ..., k trajectory-selection pairs of (2.3) such that uj =wj(T ), j = 1, ..., k. We note that β > 0 can be take smallenough such that ||wj(0)|| ≤ 1, j = 1, ..., k.

For s = (s1, ..., sk) ∈ Rk we define

w(t, s) =k∑j=1

sjwj(t), g(t, s) =k∑j=1

sjgj(t).

Obviously, w(., s) ∈ SP , ∀s ∈ Σk.

Taking into account the definition of C0, for every ε > 0there exists a continuous mapping oε : Σk → Rn such that

z(0) + εw(0, s) + oε(s) ∈ X0, (3.4)

limε→0+

maxs∈Σk

||oε(s)||ε

= 0. (3.5)

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Define

pε(s)(t) :=1

εd(g(t, s), F (t, z(t) + εw(t, s))− f(t)),

q(t) :=k∑j=1

[||gj(t)||+ L(t)||wj(t)||], t ∈ I.

Then, for every s ∈ Σk one has

pε(s)(t) ≤ |g(t, s)|+ 1εdH(0n, F (t, z(t) + εw(t, s))−

f(t)) ≤ |g(t, s)|+ 1εdH(F (t, z(t)), F (t, z(t) + εw(t, s)))

≤ |g(t, s)|+ L(t)|w(t, s)| ≤ q(t).(3.6)

Next, if s1, s2 ∈ Σk one has

|pε(s1)(t)− pε(s2)(t)| ≤ |g(t, s1)− g(t, s2)|+1εdH(F (t, z(t) + εw(t, s1)), F (t, z(t) + εw(t, s2)))≤ ||s1 − s2||.maxj=1,k[|gj(t)|+ L(t)|wj(t)|],

thus pε(.)(t) is Lipschitz with a Lipschitz constant not depend-ing on ε.

On the other hand, from (2.7) it follows that

limε→0

pε(s)(t) = 0 a.e. (I), ∀s ∈ Σk

and hence

limε→0+

maxs∈Σk

pε(s)(t) = 0 a.e. (I). (3.7)

Therefore, from (3.6), (3.7) and Lebesgue dominated con-vergence theorem we obtain

limε→0+

∫ T

0

maxs∈Σk

pε(s)(t)dt = 0. (3.8)

By (3.4), (3.5), (3.8) and the upper semicontinuity of theClarke generalized Jacobian we can find ε0, e0 > 0 such that

maxs∈Σk

||oε0(s)||ε0

+

∫ T

0

maxs∈Σk

pε0(s)(t)dt ≤ β1

28l2, (3.9)

ε0w(T, s) ≤ e0

2∀s ∈ Σk. (3.10)

If we define

y(s)(t) := z(t) + ε0w(t, s), g(s)(t) := f(t) + ε0g(t, s),

a(s) := z(0) + ε0w(0, s) + oε0(s), s ∈ Rk,

then we apply Theorem 2.7 and we find that there exists thecontinuous function x(.) : Σk → C(I,Rn) such that for anys ∈ Σk the function x(s)(.) is a C0-solution of the differentialinclusion x′ ∈ Ax + F (t, x), x(0) = a(s) ∀s ∈ Σk and onehas

||x(s)(T )− y(s)(T )|| ≤ ε0β1

26l∀s ∈ Σk. (3.11)

We define

h0(x) :=

∫Rn

h(x− by)χ(y)dy, x ∈ Rn,

φ(s) := h0(z(T ) + ε0w(T, s)),

where χ(.) : Rn → [0, 1] is a C∞ function with thesupport contained in BRn that satisfies

∫Rn χ(y)dy = 1 and

b = min e02 ,ε0β1

26l .Therefore h0(.) is of class C∞ and verifies

||h(x)− h0(x)|| ≤ lb, (3.12)

h′0(x) =

∫Rn

h′(x− by)χ(y)dy. (3.13)

In particular

h′0(x) ∈ coh′(u); ||u− x|| ≤ b, h′(u) exists,

φ′(s)µ = h′0(z(T ) + ε0w(T, s))ε0w(T, µ) ∀µ ∈ Σk.

Using again the upper semicontinuity of Clarke’s general-ized Jacobian we obtain

d(h′0(z(T ) + ε0w(T, s)), ∂h(z(T ))) ≤ supd(h′0(u),∂h(z(T ))); |u− z(t)| ≤ |u− (z(T ) + ε0w(T, s))|+|ε0w(t, s)| ≤ e0, h′(u) exists < ρ1.

The last inequality with (3.3) gives

ε0β1BRm ⊂ φ′(s)Σk ∀s ∈ Σk.

Finally, for s ∈ Σk, we put ψ(s) = h(x(s)(T )).

Obviously, ψ(.) is continuous and from (3.11), (3.12),(3.13) one has

||ψ(s)− φ(s)|| = ||h(x(s)(T ))− h0(y(s)(T ))|| ≤||h(x(s)(T ))− h(y(s)(T ))||+ ||h(y(s)(T ))−h0(y(s)(T ))|| ≤ l|x(s)(T )− y(s)(T )|+ lb ≤ε0β1

64 + ε0β1

64 = ε0β1

32 .

We apply Lemma 2.11 and we find that

h(x(0k)(T )) +ε0β1

16BRm ⊂ ψ(Σk) ⊂ h(RF (T )).

On the other hand, ||h(z(T ))− h(x(0k)(T ))|| ≤ ε0β1

64 , so wehave h(z(T )) ∈ int(RF (T )) and the proof is complete.

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