CC BY
1y(e)\\ = Hte.si.a»)!! < \ls\\Xi - x2\\2 < .
To achieve this, it is necessary to choose a value of e such that it satisfies the inequality
f v v _
s ^ to = min { —— ,—,£ 14Lr'2k'
Then, from Lemma 1 with parameters
||xi - x2||2
At = ö' P =
2' ' 8r
it follows that there exists x* € B(x0, p) such that F(x*,e) = y(e).
Since BX(0, r) is Hilbert ball, it is strongly convex and the inclusion B(x0,p) C BX(0, r) is true, therefore x* € BX(0, r). So, the point
y{s) = \(F{Xl,s) + F{X2,S))
is contained within the image of the ball F(BX(0, r),e( for all e s) e0 and x1,x2 € BX(0, r). Due to the closeness, for all e S e0, the image of the ball F(BX(0, r), e) is convex. □
4. On the properties of the solutions of quasilinear systems
In this section we investigate the solutions of (2.1) to verify the applicability of the previous results, in particular Theorem 1.
By X(t,r) we denote the Cauchy matrix of the linear system x(t) = A(t)x(t). This matrix is the solution of the following equation
^f^ = A(t)X(T,r), X(r,r)=F
If x(-, e,u(-)) is the solution of (2.1), produced by control u(-) and initial condition x0, it satisfies the next integral equation
T
x(T,e,u(■ )) = X(T,t0)x0 + yX(T,t^Bu(t) + e^x(r,e,u(-)),^ dr
to
T T
= X (T,t0)x0 + J X (T, t )B(i)u(r) dr + e J X (T,t )^x(r,e,u( ■ )),r) dr.
to to
Let us define the mapping F : _Bl2(0,]u) x [0,e] —> Mra by the equality F(u(-),e) = x(T,e,u(-)), where x(T, e, u(■ )) is the solution of (2.1) at moment T generated by the control u(■ ) and the small parameter e.
In order to use the results from the previous sections, we now rewrite the mapping F as
F (u( ■ ), e) = a0 + A0u( ■ ) + eA1(u( ■ ),e),
where a® = X(T,0)xo, the linear map Aq : £>l2(0,^) h> R" is defined by
T
Aqu(-) = y x(T, r)B(t)u(r) dr
to
and nonlinear map A\ : B^,2(0,Ji) x [0,e] —> Rra is defined by
T
)"\x(r,e,u(
to
T
Al(u(-),e) = J X (T,r )f (x(r,e,u(-)),r) dr.
(4.1)
Reachable set G(T,^,e) of the quasilinear system (2.1) is the image under mapping F of the ball Bu (0,
G(T,p,e) = F (BL2 (0, y),e).
Assertion 1. Assume the Assumption 1 is fulfilled. Then, for all s € [0,e], the reachable set G(T,^,e) is closed.
Proof. The proof is based on the equicontinuity of trajectories, the uniform boundedness of the set of trajectories, and the weak compactness of the ball BL2 (0,^) (see, for example [11]). □
To apply Theorem 1 to the mapping F, we must demonstrate that Assumption 2 holds for Ai, defined in equation (4.1).
Lemma 2. Assume Assumption 1 to be fulfilled. Then, for alls € [0,e], there exists a constant Lx(e) > 0, such that for any ui(-) € BL2(0,^), i = 1,2 and t € [t0,T],
||xi(t) - X2(t)|| < Lx(e)||ui(-) - U2(-)^L2, where Xi(t) = x(t,e,Ui(-)), i = 1,2. Furthermore, Lx(e) ^ Lx(e).
Proof. Since Xi(t) € D for all t € [t0,T], from Assumption 1, we have ||f (xi(t),t) - f (X2(t),t)| < Lf ||Xi(t) - X2(t)||. From the integral identities
xi
t t t (t) = xo + J A(r)xi(r) dr + J B(r)ui(r) dr + e J f (Xi(r),r) dr (4.2)
to to to
we get
||X1 (t) - X2(t)y <
A(r){xi(r) - X2(r)) dr
to
+
B(r)(ui(r) - U2(r)) dr
to
J f (xi(r),r) - f (x2(r),r)) dr
to
t
^J (kA + Lfe) ||xi(r) - x2(r)|| dr + kjui(■) - U2(-)|l2•
to
t
t
t
Here,
ku = 4 /(T — ¿0) max ||B(r)||, kA = max ||A(r)
T e[to,t] T e[to,t]
From the Grownwall inequality we have
||x1(t) — x2(t)| S Lx(e)|u1(■ ) — U2(■ )|l2,
where
Lx(e) = ku exp ((kA + L/e)(T — ¿0)). Note, that Lx{e) ^ Lx(e). □
Introduce the mapping F : [t0,T] x [0,e] x BL2(0,~jJ) ->• Rra,
F (r, e, u( ■ )) = x(r, e, u( ■ )),
where x(r, e, u(■ )) is a solution of (2.1) at moment r generated by the control u( ■ ) the small parameter t. The derivative of F in u(-), f' : £>L2(0,7t) —> is the solution of the linearized system as it was shown in [11]
F(r,e,u( ■ ))M ■ ) = 5x(r),
where ¿x(r) is a solution of the the system (2.1) linearized along (u( ■ ),x(-,e, u(■ )), corresponding to the control ■ ) and zero initial condition:
5x = A(t, e, u{-))5x + B(t)8u(t), O^t^r, 5x{ 0) = 0, (4.3)
where
0f (x(t,e,u( ■ )),t)
A(t,e,u{-)) = A(t) + t-
dx
Lemma 3. Suppose Assumption 1 to be fulfilled. There exists a constant Lu(e) > 0, such that
for any e € [0,e], ih(-) € £>l2(0,^) and r € [to,?1].
\\F'(r, s, «!(•)) -F'(w>2(-))II < L„(e)||«i(-) -«2(-)IIL2>
where i = 1,2.
Proof. The solution of (4.3) has the form
T
6x(t,£,ui{-),6u{-)) = J X(t, s, e, Ui(-))B(s)6u(s) ds, (4.4)
to
where X(r, s,e, u( ■ )) is fundamental matrix of system (4.3), and it satisfies the equation
= -a(s,£,u(-))tx(t,s,£,u(-)), x(t,t,£,u(-)) =F
It is well-known (for example, it follows from the proof of Theorem 3 in [7]), that there exists kx > 0 such that
\\X(T,s,£,u(-))\\ < kx, T £ [to,T], s € [to,T]
for all u(-) £ B]i2(0,fi) and sufficiently small t. For the sake of brevity, we use the notation Ai(t) = A(t,e,Ui(•)) and Xi(t,s) = X(t,s,e,Ui(-)). Under Assumption 1 and using Lemma 2 we can obtain the estimate
T
I IPMs) -A2(s)\\ds < LA\\Ul(-) -m2(-)IIl2.
to
Here La > 0 does not depend on u1( ■ ), u2(■ ), t and e. Since,
a _____T _____ _ _
— (Xi(i, s) - X2(t, s)) = -a; (t) (Xi(i, s) - X2(t, s)) + (a2(t) - A1(t))TX2(t, s), t£ [s, r]
we get the following formula
T
Xi{t,s)-X2{t,s) = J Y(t, s) (A2{t) — ~A\(t))TX2(t, s)dt.
s
Here Y(t, s) is a fundamental matrix of the system
y = -Mt)y.
Like Xi(r,s), this matrix is also bounded: there exists ky > 0 such that
||Y(t, s)|| < ky, t,s € [to,T]
for all u(■ ) € B(0,v). We get
||Xi(r,S) -X2(r,S)|| < LX\\U1(-) -u2(-)\\U,
where
Lx = kyLAkx(T - to).
Hence from (4.4) it follows the statement of the lemma and Lu(e) = Lx(e)T. □
Now we will claim Frechet differentiability of the mapping Ai(u( ■ ),e) in u( ■ ). Let us choose arbitrary u(-) £ B]i2(0,fi) and Su(-), such that ||^/,(-)||l2 ~ and consider
Ai(u( ■ ) + 5u(■ ),e) - Ai(u( ■ ),e)
t
= i X (T, r ) [f(x(r,s,u( ■ )+ Su( ■ )),r) - f(x(r,s,u( ■ )),r)
(4-5)
dr.
to
Here we should study the difference between solutions of (2.1), produced by u(■ ) and u(■ )+Su(■ ). From (4.2) it follows
t
x(t,,M■) + M )) - )) = / A(T) [xCr,S,u(■) + M■)) - x(w<■))] dr
to
t t + / B(T)MT) dr + ej [f ^M •) + M )),r) - f (x(T,eM- )),r)] dr.
to to
(4-6)
Let y € Rn and h € Rn be chosen such that the inclusions y € D and y + h € D are valid. Then, for all t € [to,T], using representation of the increment of a function through the integral over a parameter, we have
f(y + h, t) - f(y, t) = ( J £ {y + £h, t) d^j h = (y, T)h + u(y, h, r),
0
where
1
0
Since D is convex, y + {h € D for all 0 S C S 1. Therefore, using Assumption 1, we can obtain the following estimate
i
Mv,h,T)\\^lfQ ll^ll
o
When y = x(r, e, u(■ )) and
h = Ax(r, e, ■ )) = x(r, e, u(■ ) + ■ )) — x(r, e, u(■ )), for all r € [t0,T] we have
f (x(r,e,u( ■ ) + 5u( ■ )),r) — f (x(r,e,u( ■ )),r) df (4.7)
= — (x(r, t, «(•)), t)Ax(t, £, £«(•)) + co(x(t, £, u(-)), Ax(t, £, Su(-)), t) ,
where (see Lemma 2)
|W(x(r,t-,t.(-)),Ax(T,t-,^(-)),T)|| < ||Ax(r,t-,M-))||2 < ^OOIIMOIIL- (4.8)
From (4.7) it follows, that w(x(r, e, u(■ )), Ax(t,e, ■ )), ■ ) is measurable, as the sum of measurable functions. Substituting (4.7) to (4.6), we obtain
Ax(t,e,ôu{-)) = / A( t, e, u(■ ))Aï(t,£,îu(■ )) dT + J B(t) dT
to to
t
j w(x(T,e,u( ■ )), Aï(t,£,îu( ■ )),t^t = &x(i) + Q(i, e, ■ )),
to
where fe(i) is the solution of system (4.3) and
t
Since (4.8) we can estimate Q(t,e,öu(■ )) from above for all t € [to,T]
Here we are going to rewrite (4.7),
f(x(r,£,u(■ ) + Su(■ )),r) - f(x(r,£,u(■ )),r)
df df
= ——- (x(t, £, «(•)), T)6x(t) + (,t(t, t, «(•)), t) t, £«(•)) + w(.t(t, t, u(-)),Ax(t, £, <5«(-))> r) .
dX dX
We can estimate the norm of residial term from above:
< lj£Ll(£)(T-to) max t) || ||5u(-)
t €[to,T ]
IL2 '
Therefore, we are able to rewrite (4.5) in form
t
A1(u(-) + 6u(-),e)-A1(u(-),e) = JX(T, r)|£ (.r(r, e, «(•)), r)fe(i) dr + o(||5u(-)ll2)-
to
This implies, that the Frechet. derivative A^ («(•), t) : £>L2(0,7t) —>■ exists and could be defined by equality
t
A'1{u{-),e)Su{-) = JX(T,t)^(x(t,e,u(-)),t)8x(t) dr
to
(4.9)
The Lipschitz continuity of Sx(■ ) was proved in Lemma 3. The derivative
df ,
dx
x(r, e, u(■ )), t
is Lipschitz continuous as a composition of Lipschitz continuous functions
df (x(t, e, ui ( ■ )),t) df(x(T,e,U2 ( ■ )), t)
dx dx
< If Lx(e) ||ui(■ ) - U2( ■ )yL2
^ If ||x{r,e,u1 ( ■ )) — x(r,e,u2( ■ ))
T e [to,T], ui(■ ),U2(■ ) e Bl2(0,y). Then the integrand in (4.9) also fulfills the Lipschitz condition for all t € [0,£] and r € [to,?1],
{x(T, £, ui(-)),t)f'(t, £, Ul(-))ÔU(-) - {x(T, £, U2(-)), t)f' (t, t, u2(-))6u(-)
< (ß ~ ß) lfLx(e) , max ||F (r, t, «(•)) || + Lu(e) max
T€[to,T] T€[to,T]
f
dx V'
x, T
||ui( ■ ) — u2( ■
and the whole derivative A'1(u(■ ),e) will be Lipschitz continuous in u( ■ ).
In order to fulfill the condition of Assumption 2, it remains to show that this derivative will be continuous in t. This is valid due to the facts that the right-hand side of system (2.1) is linear in the parameter t and the matrix A(t,e,u(-)) of the linearized system (4.3) depends continuously on e.
Thus, the mapping A1(u( ■ ),e) defined in (4.1) fulfills the condition of Assumption 2 and we are able to formulate the main result of this paper in the following theorem
Theorem 2. Assume the conditions of Assumption 1 are satisfied, then there exists a positive value £0 such that the reachable sets G(T, of the quasilinear system (2.1) are convex for all £ < £o-
Proof. The statement's validity can be confirmed by applying Theorem 1 to the mapping F, given that Lipschitz continuity of and closeness of G(T, £) (Assertion 1) were previously established. □
Remark 1. In the article [2], E.G. Albrecht investigates the support functions of reachable sets for quasilinear systems with integral constraints. The paper defines conditions under which the support functions of reachable sets have continuous dependence on parameter. The author also noted that the continuous dependence of the reachable set on the parameter implies its convexity for small values of parameter. However, no proof of this fact was provided. Furthermore, continuity of reachable sets alone was not sufficient to prove it.
5. Examples
In this section, we present the results of numerical experiments that are intended to illustrate the application of the Theorems 1 and 2.
Example 1. First system under study is Duffing oscillator. We deal with equations
x'i = x2, x2 = —x1 — 10exi + u, 0 ^ t ^ 2 (5.1)
describing the motion of a non-linear elastic spring under the influence of an external force u. The impact of the nonlinear elastic force term is determined by the small parameter e > 0. The initial state is x1(0) = x2(0) = 0, and the control is bounded by
2
J u2dt ^ 1. (5.2)
0
When e = 0, the equations (5.1) describe a linear system with the matrices
A = (—1 1) ' B = (1
The nonlinear term comprises of a small parameter and the function f (x) = [—10x3 ;0]. Condition (2.2) is not fulfilled for this nonlinear term. However, we can use estimates obtained in paper [26] to show, that all the trajectories of the system (5.1) corresponding to admissible controls and zero initial state are lying in a compact set D. We set
, . 5 4 1 2 1 2
v£(t,x) = -ex i + -X! + -x2
and calculate the time derivative d
dtVe(t; x(t)) = Vv£(t, x(t)) (Ax(t) + Bu(t) + ef(x(t))) = x2(t)u(t). (5.3)
For each £ ^ 0 and each control u( • ) satisfied (5.2), there exists t > 0, such that the solution of (5.1) generated by this control u( • ) and by zero initial state is defined on time interval [0, t]. Let us integrate (5.3) from 0 to t. We have
T 2 1/2 T 1/2 T 1/2
v£(t,x(t)) = J X2(t)u(t) dt ^ (^J u2(t) dt^j ^ J xl(t) d?j ^ y/2^ J v£(t,x(t)) d?j .
ooo o
Applying comparison theorem to this inequality, one can obtain, that (t, x(t)) ^ t and, therefore, ||x(t)||2 ^ 2t. Using well-known technique, we could conclude that any solution (5.1) generated by a control u( • ) E BL2 (0,1) and zero initial state, could be continued to time interval [0, 2] and it will belong to the convex set D — Br«. (0, 2).
The Assumption 1 are fulfilled: the pair (A, B) is a constant; the function f is continuous and continuously differentiable; also, the function f and its derivative df/dx satisfy the Lipschitz condition on the set D.
Therefore, the requirements of Theorem 2 are fulfilled for system (5.1), and the corresponding reachable sets should be convex for small parameter values. This is evident in Fig. 1, which demonstrates the constructed reachable sets G (T, using numerical Monte-Carlo based technique [24, 25].
It can be seen that sets Gao1 (1,1) and Go1 (1,1) are close to set Go (1,1) constructed for the linear system. One can also see that the sets become non-convex as the parameter £ increases.
Figure 2. The reachable sets of system (5.4).
Example 2. Second system under study is
'0 1 0\ /xA /cos x3 - x2\ /0'
0 0 1 x2 + £ sin X3 - x3 + 0 | u.
0 0 0 VxJ V 0 / \1
When £ — 0, the equations (5.4) describe a linear system with matrices
(5.4)
A
010 001 001
B
and when £ =1, they describe a unicycle. The nonlinear term comprises of a small parameter and the function
f (x) =
cos x3 sin x3 0
The initial state is zero x1 (0) — x2 (0) — x3 (0), the constraints on the controls are the same as in the first example, but we will consider this system on the time interval 0 ^ t ^ 1.
Similar to the previous example, the conditions of Assumption 1 are satisfied, allowing the application of Theorem 2. Fig. 2 displays the projections in the plane (x1 ,x2) of the numerically constructed reachable sets G(T, for the system (5.4).
It can be seen that projections of sets G°'°o1 (1,1) and Go>o1 (1,1) are close to projection of set Go(1,1) constructed for the linear system. One can also see that the projections of sets become non-convex as the parameter £ increases.
Acknowledgements
The author is grateful to the anonymous reviewers for their careful scrutiny of the manuscript and their constructive feedback, which greatly contributed to its revision.
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