introduction to the mathematical theory of control, lecture 3motta/lectures valona 2017...control...

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Introduction to the Mathematical Theory of Control,Lecture 3

Monica Motta

Dipartimento di MatematicaUniversità di Padova

Valona, September 13, 2017

M. Motta (Padua Un.) Control Theory Valona, September 13, 2017 1 / 16

Table of contents

1 The input-output map: differentiability w.r.t. the control-conclusion

2 Closure of the set of solutions

3 Reachable sets


Differentiability w.r.t. the control

The continuous dependence of trajectories on the control function u is abasic result. However, in the analysis of optimal control problems, strongerregularity properties are needed.

Theorem 1 (Differentiability w.r.t. the control).Assume (H) and let f be defined on ⌦⇥ V, with V open neighborhood ofU, and continuously differentiable w.r.t. u. Let u(·) 2 U be a control withx(·, u) defined on [0,T ]. Then, for every bounded measurable � u(·) andevery t 2 [0,T ], the map " 7! x(t , u + "� u) is differentiable and itsderivative at " = 0+ is

dd" x(t , u + "� u)|

"=0+=

R t0 M(t , s)Duf (s, x(s, u), u(s)) ·� u(s) ds.

M is the matrix fundamental solution for the linearized problem

v(t) = Dxf (t , x(t , u), u(t)) · v(t).

Proof.M. Motta (Padua Un.) Control Theory Valona, September 13, 2017 3 / 16

Equivalent formulation

As discussed yesterday,

z(t) :=Z t

0M(t , s)Duf (s, x(s, u), u(s)) ·� u(s) ds

can be equivalent represented as the solution of the linear system(

z(t) = Dxf (t , x(t , u), u(t)) · z(t) + Duf (t , x(t , u), u(t)) ·� u(t),

z(0) = 0

where u(·) and x(·, u) are given, so thatt 7! A(t) := Dxf (t , x(t , u), u(t)) is a measurable n ⇥ n matrix,t 7! B(t) := Duf (t , x(t , u), u(t)) is a measurable n ⇥ m matrix.

Thus the thesis is:

dd" x(t , u + "� u)|

"=0+= z(t)


Closure of the set of solutions

Consider a sequence of admissible controls u⌫ 2 U , and assume that thecorresponding solutions x(·, u⌫) of the Cauchy problem

x(t) = f (t , x , u), x(0) = x (1)

converge to x(·) uniformly on [0,T ]. Our main concern is whether this limittrajectory is a solution of the original control system. In general, this maynot be the case, since the controls u⌫(·) might have a highly oscillatorybehavior and NOT converge in L1.

Example 2.Consider the system on R:

x(t) = u(t), x(0) = 0, u(t) 2 {�1, 1}.

Let x⌫(·) be the trajectories associated to the controls (k integers)

u⌫(t) = 1 if k⇡/⌫ t (k + 1)⇡/⌫; u⌫(t) = �1 otherwise.


The trajectoriesx⌫(·) convergeuniformly tox(·) ⌘ 0 on R.However, x(·) ⌘ 0is NOT a solution.

The closure of the set of trajectories is best studied within the frameworkof differential inclusions.

Theorem 3 (Filippov’s Thm. on the closure of the set oftrajectories).Assume that the multifunction (t , x) 7! F (t , x) is Hausdorff continuouson R⇥ Rn with compact convex values. Then the set of trajectories of

x(t) 2 F (t , x(t)) a.e. t 2 [0,T ] (2)

is closed in C0([0,T ],Rn).


Continuity concepts for multifunctions

Let X , Y be metric spaces. Let F : X ! P(Y ) be a multifunction withcompact values.

The continuity of a function f : X ! Y at x 2 X can be stated in twoequivalent ways:

using sequences: for any x⌫ ! x one has f (x⌫) ! f (x)

in a topological way: for every " > 0 there exists � > 0 such that forany x 0 2 BX (x , �), one has f (x 0) 2 BY (f (x), ").


Extended to multifunctions, these two notions are NO MOREEQUIVALENT, and give rise, respectively, to the definitions

Definition 4.A multifunction F : X ! P(Y ) with compact, nonempty values is said tobe

lower semicontinuous (lsc) at x if and only if for any y 2 F (x) and forany x⌫ ! x , there exists y⌫ 2 F (x⌫) converging to y ;

upper semicontinuous (usc) at x if and only if for every " > 0 thereexists � > 0 such that for any x 0 2 BX (x , �), one hasF (x 0) 2 BY (F (x), ").

We call F continuous at x if it is both lsc and usc at x .


Examples of lsc, usc multifunctions

Let us consider the multifunctions F1, F2 : R ! P(R) defined as

F1(x) =

([�1, 1] if x 6= 0

{0} if x = 0, F2(x) =

([�1, 1] if x = 0

{0} if x 6= 0

At x = 0, we have


Picture of the graph of a multifunction F : R ! P(R) continuous on itsdomain.


It can be proven that a multifunction F : X ! P(Y ) with compact,nonempty values

is continuous at x if and only if it is Hausdorff-continuous, that is, ifand only if

limx 0!x

dH(F (x 0),F (x)) = 0

has closed graph if and only if F is upper semicontinuous.

Multifunctions are known also as set-valued maps. They are studied in theframework of Set-Valued Analysis (starting from the ’30s)


Go back to the proof of Filippov’s Thm.. It makes use of the followingproperty of convex sets (Hahn-Banach Thm., second geometric form).

Lemma 5 (Separation of convex sets).Let K , K 0 ⇢ Rn be disjoint, closed, convex sets, with K compact. Thenthey can be strictly separated by a hyperplane. More precisely, there exists" > 0 and a unit row-vector p 2 Rn such that

minv2K

p · v � supv 02K 0

p · v 0 + ".


Lemma 6.If f : R⇥ Rn ⇥ U ! Rn is continuous and U ⇢ Rm is compact, then themultifunction

(t , x) 7! F (t , x) := {f (t , x , u) : u 2 U}

has compact values and is Hausdorff- continuous on R⇥ Rn.

Proof.Using this lemma and the equivalence Thm. between control systems anddifferential inclusions, the previous result yields

Corollary 7.Let the basic assumptions (H) hold. Let x⌫(·) be a sequence of solutionsto (1) converging to x(·) uniformly on [0,T ]. If

the graph {(t , x(t)) : t 2 [0,T ]} is entirely contained in ⌦;

all sets of velocities F (t , x) = {f (t , x , u) : u 2 U} are convex,

then x(·) is also a trajectory of the control system (1).


In the above results, the key assumption is the convexity of the sets ofvelocities F (t , x) ⇢ Rn, NOT the convexity of the set of controls U ⇢ Rm.

If f (t , x , u) is affine in u, that is, f (t , x , u) = g(t , x) + B(t , x)u(B n ⇥ m matrix), then U convex implies F (t , x) convex.

In general, if U is convex but f is non-linear in u, the sets F (t , x) maynot be convex.

Example 8.

Consider the control system on R2

(x1, x2) = (u, 1 � u2) u 2 U := [�1, 1].

Here U is convex. However, consider the sequence of rapidly oscillatingcontrols u⌫ of the previous example. Starting from the origin at time t = 0,the corresponding trajectories t 7! x⌫(t) = x(t , u⌫) converge to the nulltrajectory x(t) = (0, 0), uniformly w.r.t. t . However, this is not a trajectoryof the system, for any control u(·). In fact the set of velocities

F (t , x) = {(y1, y2) : y2 = 1 � y21 , y1 2 [�1, 1]} is not convex!


Reachable sets

Let us consider a control system whose dynamics is independent of time:

x(t) = f (x(t), u(t)), x(0) = x , u(·) 2 U . (3)

The reachable set R(⌧, x) at time ⌧ starting from x , is then defined as

R(⌧, x) := {x(⌧) : x(·) solution of (3)}. (4)


The next theorem establishes the closure of the reachable sets, under asuitable convexity assumption. This closure property will be of greatimportance, providing the existence of optimal controls.

Theorem 9 (Closure of reachable sets).Assume (H). If

P.1 the graphs of all solutions of (3) are contained in some compact setK ⇢ ⌦ for t 2 [0,T ].

P.2 all sets of velocities F (x) := {f (x , u) : u 2 U} are convex

then, for every ⌧ 2 [0,T ], the reachable set R(⌧, x) is compact.

Remark. More generally, an analogous result holds true for the reachableset at time ⌧ , starting from points in some set K 0, that is

R(⌧,K 0) := {x(⌧) : x(·) solution of (3) for some x 2 K 0}.

Moreover, everything can be extended to t-dependent data.M. Motta (Padua Un.) Control Theory Valona, September 13, 2017 16 / 16

Nuova sezione 2 Pagina 11

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