Some sufficient conditions for controllability of nonlinear systems described by differential equation ˙x = f (t, x(t), u(t)) are given.

CONTROLLABILITY THEOREM FOR NONLINEAR DYNAMICAL SYSTEMS

MichaÃl Kisielewicz Institute of Mathematics University of Zielona G´ora

Podg´orna 50, 65–246 Zielona G´ora, Poland

Abstract

Some sufficient conditions for controllability of nonlinear systems described by differential equation ˙x = f (t, x(t), u(t)) are given.

Keywords: differential equation, differential inclusions, controllabil- ity, boundary value problem.

2000 Mathematics Subject Classification: 54C65, 54C60.

1. Introduction

It was proved in the author’s paper [2] that for a given multifunction F : [0, T ] × IR ⁿ → Conv(IR ⁿ ) satisfying the usual Carath´eodory type condition, the boundary value problem

 



˙x(t) ∈ F (t, x(t)) for a.e. t ∈ [0, T ] x(0) = x ₀ , x(T ) = x ₁

(1)

has for every x ₁ , x ₀ ∈ IR ⁿ at least one solution if and only if there is a nonempty weakly compact set Λ ⊂ L ^F _x

_−x

⊂ L([0, T ], IR ⁿ ) such that

x ₁ − x ₀ ∈ Z _T

0 F

µ t, x ₀ +

Z _t

0 v(τ )dτ

¶

dt

(2)

for every v ∈ Λ, where L ^F _x

_−x

denotes the set of all v ∈ L([0, T ], IR ⁿ ) such that |v(t)| ≤ sup _x∈IR

kF (t, x)k a.e. on [0, T ] and x ₁ − x ₀ = ^R ₀ ^T v(t)dt.

Furthermore, it was proved in [2] that (2) is equivalent to the inequality:

< p, x ₁ − x ₀ > ≤ X N i=1

µ(E _i ) sup

t∈E

h _{F (t,x}

0

+ R

0

v(τ )dτ )dt (p) (3)

for every measurable partition {E ₁ , . . . , E _N } to [0, T ], v ∈ Λ and p ∈ S ⁿ = {x ∈ IR ⁿ : kxk = 1}, where h _C denotes the support function of a set C ⊂ IR ⁿ . We shall use the above equivalences to define some sufficient conditions for the existence of a pair (x, u) ∈ AC([0, T ], IR ⁿ ) × M([0, T ], IR ^m ), such that

 



˙x(t) = f (t, x(t), u(t)) for a.e. t ∈ [0, T ]

x(0) = x ₀ , x(T ) = x ₁ , u(t) ∈ U (t) for a.e. t ∈ [0, T ], (4)

where f : [0, T ] × IR ⁿ × IR ^m → IR ⁿ satisfies the Carath´eodory conditions, x ₁ , x ₀ ∈ IR ⁿ and U : [0, T ] → Conv(IR ^m ) is a measurable set valued map- ping. As usual AC([0, T ], IR ⁿ ) and M([0, T ], IR ^m ) denote the spaces of all absolutely continuous and of all measurable, respectively, functions on [0, T ].

We denote by Comp(IR ^k ) and Conv(IR ^k ) spaces of all nonempty compact and nonempty compact convex subsets of IR ^k .

2. Controllability theorem

We begin with the following lemmas.

Lemma 1. Let A ∈ IR ⁿ and G : [0, T ] → Comp(IR ⁿ ) be measurable and bounded. Suppose for every measurable partition {E ₁ , . . . , E _N } to [0, T ] and i = 1, . . . , N there exists (t _i , x _i ) ∈ Graph(G|E _i ) such that

< p, A > ≤ X N i=1

µ(E _i ) < p, x _i >

(5)

for every p ∈ S ⁿ , where G|E _i denotes the restriction of G to the set E _i . Then for every measurable partition {E ₁ , . . . , E _N } to [0, T ] and p ∈ S ⁿ one has

< p, A > ≤ X N i=1

µ(E _i ) sup

t∈E

h _G(t) (p).

(6)

P roof. The proof follows immediately from the inequalities

< p, A > ≤ X N i=1

µ(E _i ) < p, x _i >

≤ X N i=1

µ(E _i )h _G(t

₎ (p) ≤ X n i=1

µ(E _i ) sup

t∈E

h _G(t) (p).

Lemma 2. Suppose F : [0, T ] × IR ⁿ → Conv(IR ⁿ ) and x ₁ , x ₀ ∈ IR ⁿ are such that the following conditions are satisfied

(i) F (·, x) is measurable for fixed x ∈ IR ⁿ , (ii) F (t, ·) is u.s.c. for fixed t ∈ [0, T ], (iii) F is bounded,

(iv) L ^F _x

_−x

6= ∅.

If furthermore for every measurable partition {E ₁ , . . . , E _N } to [0, T ], v ∈ L ^F _x

_−x

and i = 1, . . . , N there exists (t _i , x _i ) ∈ Graph(F ◦ v|E _i ), such that

< p, x ₁ − x ₀ > ≤ X N i=1

µ(E _i ) < p, x _i >

(7)

for p ∈ S ⁿ , where (F ◦ v)(t) = F (t, x ₀ + ^R ₀ ^t v(τ )dτ ), then there exists x ∈ AC([0, T ], IR ⁿ ) such that conditions (1) are satisfied.

P roof. The proof follows immediately from Theorem 1 given in [2] and Lemma 1.

Corollary 1. Suppose F : [0, T ] × IR ⁿ → Conv(IR ⁿ ) and x ₁ , x ₀ ∈ IR ⁿ

are such that conditions (i) – (iv) of Lemma 2 are satisfied. If for every

measurable partition {E ₁ , . . . , E _N } to [0, T ], v ∈ L ^F _x

_−x

and i = 1, . . . , N

there exists t _i ∈ E _i such that

x ₁ − x ₀

T ∈ ^\

v∈L

F µ

t _i , x ₀ + Z _t

0 v(τ )dτ

¶

for i = 1, . . . , N

then there is x ∈ AC([0, T ], IR ⁿ ) such that conditions (1) are satisfied.

Indeed, for every i = 1, . . . , N and v ∈ L ^F _x

_−x

there is x _i ∈ F (t _i , x ₀ + R _t

0 v(τ )dτ ) such that

µ(E _i ) < p, x ₁ − x ₀

T > ≤ µ(E _i ) < p, x _i > f or p ∈ S ⁿ . Thus (7) is satisfied.

Now we obtain the following controllability theorems.

Theorem 3. Suppose f : [0, T ] × IR ⁿ × IR ^m → IR ⁿ and x ₁ , x ₀ ∈ IR ⁿ are such that:

(i) f (·, x, u) is measurable for fixed (x, u) ∈ IR ⁿ × IR ^m , (ii) f (t, ·, ·) is continuous for fixed t ∈ [0, T ],

(iii) f (t, x, ·) is affine for fixed (t, x) ∈ [0, T ] × IR ⁿ , (iv) f is bounded,

(v) L ^f _x

_−x

6= ∅.

Assume furthermore that U : [0, T ] → Conv(IR ⁿ ) is a measurable and bounded set-valued mapping such that for every measurable partition {E ₁ , . . . , E _N } to [0, T ], v ∈ L ^f _x

_−x

, and i = 1, . . . , N there is (t _i , u _i ) ∈ Graph(U |E _i ) such that

< p, x ₁ − x ₀ > ≤ X N i=1

µ(E _i ) < p, f µ

t _i , x ₀ + Z _t

0 v(τ )dτ, u _i

¶

>

(8)

for p ∈ S ⁿ . Then there exists a pair (x, u) ∈ AC([0, T ], IR ⁿ )×M([0, T ], IR ^m ) such that conditions (4) are satisfied.

P roof. Let us observe ([1], Theorem 1) that there exists a sequence of measurable functions g _n : [0, T ] → IR ^m such that U (t) = co ^{S ∞} _n=1 {g _n (t)}.

Then, for a fixed x ∈ IR ⁿ one has

F (t, x) = f (t, x, U (t)) = co ⁴ [ ∞ n=1

{f (t, x, g _n (t))}

which implies the measurability of F (·, x) for a fixed x ∈ IR ⁿ and F (t, x) ∈ Conv(IR ⁿ ) for every (t, x) ∈ [0, T ] × IR ⁿ . Finally ([3], Proposition II.2.5), F (t, ·) is continuous for a fixed t ∈ [0, T ]. It is also clear that F is bounded and L ^F _x

_−x

6= ∅. For every measurable partition {E ₁ , . . . , E _N } to [0, T ], v ∈ L ^F _x

_−x

, and i = 1, . . . , N there is x _i = f (t _i , x ₀ + ^R ₀ ^t

v(τ )dτ, u _i ) ∈ F (t _i , x ₀ + R _t

0 v(τ )dτ ) such that for p ∈ S ⁿ the inequality (7) is satisfied. Hence, by Lemma 2, there is an x ∈ AC([0, T ], IR ⁿ ) such that conditions (1) are satisfied. In particular, we have ˙x(t) ∈ F (t, x(t)) for a.e. [0, T ]. Thus ([3], Theorem II.3.12) there is u ∈ M([0, T ], IR ^m ) such that u(t) ∈ U (t) and

˙x(t) = f (t, x(t), u(t)) for a.e. t ∈ [0, T ].

Immediately from the above theorem we obtain the following.

Theorem 4. Suppose f : [0, T ] × IR ⁿ × IR ^m → IR ⁿ and x ₁ , x ₀ ∈ IR ⁿ are such that conditions (i) – (v) of Theorem 3 are satisfied and let U : [0, T ] → Conv(IR ^m ) be bounded and measurable. Let f (t, x, u) = α(t, x)+β(t, x)·u for (t, x) ∈ [0, T ] × IR ⁿ and u ∈ IR ^m , where β(t, x) is n × m-matrix. If for every measurable partition {E ₁ , . . . , E _N } to [0, T ], v ∈ L ^f _x

_−x

and i = 1, . . . , N there is t _i ∈ E _i such that.

x ₁ − x ₀

T − α

µ

t _i , x ₀ + Z _t

0 v(τ )dτ

¶

∈ β µ

t _i , x ₀ + Z _t

0 v(τ )dτ

¶

· U (t _i ) (9)

then there exists (x, u) ∈ (AC([0, T ], IR ⁿ × M([0, T ], IR ^m ) such that condi- tions (4) are satisfied.

P roof. Let us observe that (9) imples that for every measurable partition {E ₁ , . . . , E _N }, of [0, T ], v ∈ L ^f _x

_−x

and i = 1, . . . , N there exists (t _i , u _i ) ∈ Graph(U |E _i ) such that

X N i=1

µ(E _i ) < p, x ₁ − x ₀

T − α

µ

t _i , x ₀ + Z _t

0 v(τ )dτ

¶

−β µ

t _i , x ₀ + Z _t

0 v(τ )dτ

¶

· u _i > ≤ 0,

for p ∈ S ⁿ which in particular implies the inequality (8) for every p ∈ S ⁿ . Thus the result follows immediately from Theorem 3.

Theorem 5. Suppose f : [0, T ] × IR ⁿ × IR ^m → IR ⁿ and x ₁ , x ₀ ∈ IR ⁿ satisfy

conditions (i) – (v) of Theorem 3 and let U : [0, T ] → Conv(IR ^m ) be bounded

and measurable and such that 0 ∈ U (t) for t ∈ [0, T ]. Let f (x, x, u) = α + β(t, x) · u for (t, x) ∈ [0, T ] × IR ⁿ and u ∈ IR ^m , where α ∈ IR ⁿ and β(t, x) is n × m-matrix. If (x ₁ − x ₀ )/T = α, then there exists (x, u) ∈ (AC([0, T ], IR ⁿ × M([0, T ], IR ^m ) such that conditions (4) are satisfied.

P roof. Let us observe that for every measurable partition {E ₁ , . . . , E _N } to [0, T ], v ∈ L ^f _x

_−x

, t _i ∈ E _i for i = 1, 2, . . . , N one has

0 ∈ β µ

t _i , x ₀ + Z _t

0 v(τ )dτ

¶

· U (t _i ).

This in particular implies that conditions (9) of Theorem 4 are satisfied.

Thus the result follows immediately from Theorem 4.

Finally, we get the following approximation theorems.

Theorem 6. Suppose f : [0, T ] × IR ⁿ × IR ^m → IR ⁿ and x ₁ , x ₀ ∈ IR ⁿ satisfy conditions (i), (ii), (iv) and (v) of Theorem 3 and let f (t, x, ·) be differentiable on O _h = {u ∈ IR ^m : kuk < h} for h > 0. If U : [0, T ] → Conv(IR ^m ) is bounded and measurable and for every measurable partition {E ₁ , . . . , E _N } to [0, T ], v ∈ L ^f _x

_−x

and i = 1, . . . , N there is t _i ∈ E _i such that

x ₁ − x ₀

T − f

µ

t _i , x ₀ + Z _t

0 v(τ )dτ, 0

¶

∈ β µ

t _i , x ₀ + Z _t

0 v(τ )dτ

¶

· U (t _i ) (10)

where β(t _i , x ₀ + ^R ₀ ^t

v(τ )dτ ) is the Jacobi matrix of f (t _i , x ₀ + ^R ₀ ^t

v(τ )dτ, ·) at u = 0, then for every ε > 0 there is δ _ε > 0 such that for every pair (x _ε , u _ε ) ∈ AC([0, T ] × IR ⁿ ) × M([0, T ], IR ^m ) satisfying conditions ˙x _ε (t) = f (t, x _ε (t), 0) + β(t, x _ε (t)) · u _ε (t), x _ε (0) = x ₀ , x _ε (T ) = x ₁ , u _ε (t) ∈ U (t) for a.e. t ∈ [0, T ] and such that ku _ε (t)k ≤ δ _ε for a.e. t ∈ [0, T ] one has k ˙ x _ε (t) − f (t, x _ε (t), u _ε (t))k ≤ ε for a.e. t ∈ [0, T ].

P roof. For every ε > 0 there is δ _ε > 0 such that

kf (t, x, u) − [f (t, x, 0) + β(t, x) · u]k ≤ ε

for every u ∈ (−h, h) such that kuk < δ _ε .

By Theorem 4 there exists (x, u) ∈ AC([0, T ], IR ⁿ ) × M([0, T ], IR ^m ) such that

 



˙x(t) = f (t, x(t), 0) + β(t, x(t)) · u(t) for a.e. t ∈ [0, T ] x(0) = x ₀ , x(T ) = x ₁ , u(t) ∈ U (t) for a.e. t ∈ [0, T ].

If there is a pair (x _ε , u _ε ) such that ku _ε (t)k ≤ δ _ε , then kf (t, x _ε (t), u _ε (t)) −

˙x _ε (t)k ≤ ε for a.e. t ∈ [0, T ].

Theorem 7. Suppose F : [0, T ] × IR ⁿ → Comp(IR ⁿ ) and x ₁ , x ₂ ∈ IR ⁿ are such that conditions (i), (iii), (iv) of Lemma 2 are satisfied. If {F (t, ·)} _{t∈[0,T ]} is uniformly equicontinuous and for every measurable partition {E ₁ , . . . , E _N } to [0, T ], v ∈ L ^F _x

_−x

and i = 1, . . . , N there exists t _i ∈ E _i such that

x ₁ − x ₀

T ∈ ^\

v∈L

F µ

t, x ₀ + Z _t

0 v(τ )dτ

¶

for i = 1, . . . , N

then for every ε > 0 there is x _ε ∈ AC([0, T ], IR ⁿ ) such that dist( ˙x _ε (t), F (t, x _ε (t))) ≤ ε for a.e. t ∈ [0, T ] and x _ε (0) = x ₀ , x _ε (T ) = x ₁ .

P roof. Let G(v) = cl _w S(F )(v) ∩ L ^F _x

_−x

, for v ∈ L ^F _x

_−x

, where cl _w denotes the weak closure in L([0, T ], IR ⁿ ) and S(F )(v) = {z ∈ L([0, T ], IR ⁿ ) : z(t) ∈ F (t, x ₀ + ^R ₀ ^t v(τ )dτ ) a.e. on [0, T ]}. Since S(F )(v) is decomposable, then ([3], Theorem III.2.3) cl _w S(F )(v) = S(coF )(v). Thus, by Theorem 1 given in [2]

there is x ∈ AC([0, T ], IR ⁿ ) such that ˙x ∈ cl _w S(F )(x) ∩ L ^F _x

_−x

, x(0) = x ₀ and x(T ) = x ₁ . Let (v _k ) ^∞ _k=1 be a sequence of S(F )(x) ∩ L ^F _x

_−x

such that v _k * ˙x as k → ∞ and let x _k (t) = x ₀ + ^R ₀ ^t v _k (τ )dτ for t ∈ [0, T ]. We get x _k → x as k → ∞ in the norm topology of C([0, T ]), IR ⁿ ). Furthermore, we have ˙x _k (t) ∈ F (t, x(t)) for k = 1, 2, . . . and a.e. t ∈ [0, T ]. Hence, in particular, we obtain

dist( ˙x _k (t), F (t, x _k (t))

≤ dist ( ˙x _k (t), F (t, x(t)))

+ sup _0≤t≤T h(F (t, x(t)), F (t, x _k (t))

for k = 1, 2, . . . and a.e. t ∈ [0, T ]. Therefore lim _k→∞ dist( ˙x _k (t), F (t, x _k (t))

= 0 uniformly for a.e. t ∈ [0, T ], which ends the proof.

References

[1] C.J. Himmelberg, Measurable relations, Fund. Math., 87 (1975), 53–72.

[2] M. Kisielewicz, Some remarks on boundary value problem for differential in- clusions, Discuss. Math. Differ. Incl., 17 (1997), 43–49.

[3] M. Kisielewicz, Differential Inclusions and Optimal Control, PWN – Kluwer Acad. Publ. (1991).

Received 1 April 2002