introduction to the mathematical theory of control, lecture 4motta/lectures valona...

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Introduction to the Mathematical Theory of Control,Lecture 4

Monica Motta

Dipartimento di MatematicaUniversità di Padova

Valona, September 14, 2017

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 1 / 32

Table of contents

1 Reachable sets

2 Chattering controls

3 Bang-bang controls

4 Existence of optimal controls


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 . (1)

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

R(⌧, x) := {x(⌧) : x(·) solution of (1)}. (2)


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 1 (Compactness of reachable sets).Assume (H). If

P.1 the graphs of all solutions of (1) are contained in some compact set

K ⇢ ⌦ 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 (1) for some x 2 K

0}.

Moreover, everything can be extended to t-dependent data.M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 4 / 32

Example 2.

Consider the system on R2

(x1, x2) = (u, x21 ) (x1, x2)(0) = 0, u(t) 2 U := {�1, 1}

Consider the sequence of rapidly oscillating controls of the previousexamples. Then, given T > 0, the corresponding trajectories convergeuniformly to (x1, x2)(·) ⌘ (0, 0), but this is NOT a trajectory of the system.

Indeed, if (x1, x2)(·) is a solution, then x1(t) 2 {�1, 1} implies x1(t) 6= 0 ata.e. time t . Hence the R(T , 0) is not closed, since

x2(T ) =R

T

0 x

21 (t) dt > 0.


Example 3.Consider the scalar system

x = u x

2x(0) = 1, u(t) 2 U := [�1, 1].

Notice that F (x) = [�1, 1]x2 has compact and convex values (linearsystem in u!). For each ⌫ � 1, define the control

u⌫(t) =

(1 t 2 [0, 1 � (1/⌫)]

0 t > 1 � (1⌫)=) x⌫(t) =

( 11�t

t 2 [0, 1 � (1/⌫)]

⌫ t > 1 � (1/⌫)

On every [0, ⌧ ] with ⌧ � 1, there is not a uniform bound on this set ofsolutions. In particular, x⌫(⌧) = ⌫, so that the reachable set is notbounded.

( As a consequence, the optimization problem of maximizing x(T ) for afixed T � 1 has supremum +1 and an optimal control does not exist!)


Proof (closure of the reachable set).

The proof of the theorem makes use of

Theorem 4 (Ascoli-Arzelá (simplified)).Let (f⌫)⌫ be a bounded, uniformly Lipschitz continuous sequence of

functions from a compact interval [a, b] to Rn

.

Then there exists a subsequence (f⌫0)⌫0 converging to some Lipschitz

continuous function f , uniformly on [a, b].

A natural question arises: are hypotheses P.1, P.2 reasonable forapplications?


A priori bounds

The next theorem establishes a sufficient condition for the validity ofhypothesis P.1 in the previous Theorem.

Theorem 5 (A priori bounds).In addition to the hypothesis (H), assume that f : [0,+1)⇥ Rn ⇥ U

satisfies

|f (t , x , u)| C(1 + |x |) 8(t , x , u) (sublinear growth condition)

Then, for every admissible control u(·), the solution to

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

is defined on [0,+1) and satisfies

|x(t , u)| e

Ct |x |+⇣

e

Ct � 1⌘.


Chattering controls

In many practical situations hypothesis P.2 does not hold, namely the setsof admissible velocities F (x) = {f (x , u) : u 2 U} are not convex, hencethe reachable sets may not be closed.

In this case one can provide a representation of the closure of thereachable set as reachable set of an auxiliary system

x(t) = f

](x , u]), u

](t) 2 U

] for a.e. t , (3)

in such a way that, if

F

](x) := {f

](x , u]) : u

] 2 U

]},

the trajectories to (3) are precisely the solutions to the differential inclusion

x(t) 2 F

](x) = co F (x) for a.e. t . (4)


Basic facts on convex sets

Given K ⇢ Rn,

co K denotes the intersection of all closed, convex sets containing K .co K is a closed, convex set.

By a Caratheodory’s Theorem, every point in co K can berepresented as a convex combination of at most n + 1 elements in K :

co K =

(nX

i=0

✓i

k

i

: (✓0, . . . , ✓n

) 2 rn

, k

i

2 K for all i

),

where

rn

:=

(✓ = (✓0, . . . , ✓n

) :nX

i=0

✓i

= 1, ✓i

� 0 for all i

).


As a consequence,

F

](x) = co F (x) =

(nX

i=0

✓i

f (x , ui

) : (✓0, . . . , ✓n

) 2 rn

, u

i

2 U for all i

).

Motivated by this representation, we define the compact set

U

] := U ⇥ . . .U ⇥rn

⇢ R(n+1)m+(n+1)

and the dynamics

f

](x , u]) = f

](x , (u0, . . . , un

, ✓0, . . . , ✓n

)) :=nX

i=0

✓i

f (x , ui

).

Generalized controls of the form u

] are called chattering controls.

In practical applications, they can be approximated by rapidly switching the control value u(t) among

the values u0(t), . . . u

n

(t), with the length of time during which u = u

i

proportional to ✓i

(t).


Theorem 6.Assume (H). If

P.1 the graphs of all solutions to

x(t) = f (x , u), x(0) = x , u(t) 2 U (5)

on [0,T ] are contained in some compact set K ⇢ ⌦,

then, for every ⌧ 2 [0,T ], the closure R(⌧, x) of the reachable set for the

system (5) coincides with the (compact) reachable set R](⌧, x) for the

chattering system

x(t) = f

](x , u]), x(0) = x , u

](t) 2 U

]. (6)

Proof.


The proof is based on the following result:

Theorem 7 (Approximation using a smaller set of controls).Assume (H). Consider a subset U

0 ⇢ U such that

co {f (x , u) : u

0 2 U} ◆ {f (t , x , u) : u 2 U}.

Then every trajectory of

x(t) = f (x , u), x(0) = x , u(t) 2 U

can be approximated by a trajectory of

x(t) = f (x , u), x(0) = x , u(t) 2 U

0

uniformly on bounded intervals.


Example 8.Consider again the system on R introduced yesterday:

x(t) = u(t), x(0) = 0, u(t) 2 U

0 = {�1, 1}.

Observe that U

0 ⇢ U := co(U 0) = [�1, 1] and

co{f (t , x , u) : u 2 U

0} = {f (t , x , u) : u 2 U}

The approximation Thm. says that any trajectory corresponding tou(t) 2 [�1, 1] can be approximated by trajectories with controls takingonly the values �1, 1.

Notice that the chattering control system is simply

x(t) = u(t), x(0) = 0, u(t) 2 U = [�1, 1]

In this case the approximation Thm. says also that any trajectory of thechattering system can be approximated by trajectories of the originalsystem.


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.

We can now observe that x(·) ⌘ 0 is NOT a solution for the original controlsystem, BUT it is a solution of the chattering control system(corresponding to u ⌘ 0).


Linear systems and bang-bang controls

If the sets of velocities F (t , x) = {f (t , x , u) : u 2 U} are not convex, thenwe have seen that the reachable sets may not be closed. A noteworthyexception occurs in the case of systems with linear dynamics (in x):

x(t) = A(t) x(t) + h(t , u(t)) u(t) 2 U, x(0) = x , (7)

where any point reachable using chattering controls can be reached alsoby trajectories of the original system

Theorem 9 (Reachable sets for linear systems).Assume that U ⇢ Rm

is compact, A(t) is an n ⇥ n matrix depending

continuously on t and h : [0,T ]⇥ U ! Rn

is continuous.

Then for every ⌧ 2 [0,T ], the reachable set R(⌧, x) for system (9) is a

compact, convex subset of Rn

. In other words,

R(⌧, x) = R](⌧, x).


Proof.

The proof requires to use the ’A priori bound Theorem’ and

Theorem 10 (Lyapunov’s Thm. on convex combinations).

Let f

0, . . . , f k 2 L

1([a, b],Rn) be integrable vector valued functions. Let

✓0, . . . , ✓k

: [a, b] ! [0, 1] be measurable weight functions such that

Pk

i=0 ✓i

(t) = 1 for every t.

Then there exist a partition of [a, b] into disjoint measurable subsets

J0, . . . , Jk

such that

Zb

a

kX

i=0

✓i

(t)f i(t)

!dt =

kX

i=0

Z

J

i

f

i(t) dt .

The right hand side can be interpreted as a new convex combination,where the coefficients are allowed to take only the two values 0 or 1.


As a special case, consider a linear system (in x and u) where theadmissible controls take values in a convex polytope with verticesw1, . . . ,wN

2 Rm

x(t) = A(t) x(t) + B(t) u(t) u(t) 2 U

] := co {w1, . . . ,wN

}.In addition, consider the system

x(t) = A(t) x(t) + B(t) u(t) u(t) 2 U := {w1, . . . ,wN

}.where the controls are allowed to take values only on the vertices of thepolytope. In this case, the admissible control functions u(·) are calledbang-bang controls. Indeed, they must be piecewise constant, jumpingbetween the extreme points of U

].

Corollary 11 (Bang-bang controls).Assume that the n ⇥ n matrix A(t) and the n ⇥ m matrix B(t) depend

continuously on time.

Then, for every initial point x(0) = x and any ⌧ > 0, the reachable sets

R](⌧, x), R(⌧, x) of the above systems are compact, convex andcoincide.


Mayer problem with T fixed

Consider the usual control system

x(t) = f (t , x(t), u(t)) u 2 U . (8)

Given T > 0, an initial state x , a set of admissible terminal conditionsS ⇢ Rn, and a cost function � : Rn ! R we consider the optimizationproblem

minu2U

�(x(T , u))

with initial and terminal constraints

x(0) = x , x(T ) 2 S.


Theorem 12 (Existence of optimal controls, 1).Assume (H). Let � : Rn ! R be continuous, S ⇢ Rn

closed, and moreover

P.1 the graphs of all solutions of (8) are contained in some compact set

K ⇢ ⌦ for t 2 [0,T ].

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

If some trajectory x(·) satisfying the constraints exists, then the Mayerproblem with T fixed:

minu2U

�(x(T , u))


x(0) = x , x(T ) 2 S,

has an optimal solution.


Mayer problem with free terminal time

Consider the control system (8) where now

U = {u(·) measurable : u(t) 2 U for every t}

Given an initial state x , a set of admissible terminal conditionsS ⇢ R⇥ Rn, and a cost function � : R⇥ Rn ! R we consider theoptimization problem

minT>0, u2U

�(T , x(T , u))


x(0) = x , (T , x(T )) 2 S.


We shall assume

(Hg) The control set U ⇢ Rm is compact, f : [0,+1)⇥ Rn ⇥ U iscontinuous in all the variables , C

1 in x and has sublinear growth, i.e.,

|f (t , x , u)| C(1 + |x |) for all (t , x , u).

Theorem 13 (Existence of optimal controls, 2).

Assume (Hg). Let � be continuous, S ⇢ [0, T ]⇥ Rn

closed, and moreover

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

If some trajectory x(·) satisfying the constraints exists, then the Mayerproblem with free final time has an optimal solution.


The proof is a typical example of the Direct Method for proving theexistence of optimal solutions. The basic steps are:

1. Construct a minimizing sequence (T⌫ , x⌫(·)).

2. Show that some subsequence converges to a pair (T ⇤, x⇤(·))

3. Prove that x

⇤(·) is an admissible trajectory on [0,T ⇤] and satisfies theappropriate initial and terminal conditions.

4. Prove that (T ⇤, x⇤(·)) attains the minimum value for the optimizationproblem.


Extensions

The maximization problem

maxT>0, u2U

(T , x(T , u))

is of course equivalent choosing � = � .

The minimization problem of Bolza

minu2U

⇢ZT

0L(t , x(t , u), u(t)) dt + �(T , x(T , u))

�

is equivalent to the Mayer problem

minu2U

{x0(T , u) + �(T , x(T , u))}

where x0(·, u) solves

x0 = L(t , x(t , u), u(t)), x0(0) = 0.


Convexity assumption

In the case of linear control systems (in x):

x(t) = A(t) x(t) + h(t , u(t)) u(t) 2 U, x(0) = x , (9)

the convexity assumption on the sets of velocities can be removed.

Theorem 14 (Existence of optimal controls for linear systems).

Let A, h, � be continuous, S ⇢ [0, T ]⇥ Rn

closed and U compact.

If some trajectory x(·) satisfying the constraints exists, then the Mayerproblem with free final time has an optimal solution.

Proof.

Moreover, if the control set is a polytope, one can choose the optimalcontrol to be bang-bang.


In the general non-convex case, one can often prove the existence of anoptimal chattering control, for the generalized optimization problemwhere F (t , x) = {f (t , x , u) : u 2 U} is replaced by co F (t , x).

TWO crucial questions thus arise:

1) is the infimum cost over chattering and original controls the same?(Gap phenomena may show up)2) If no gap occurs, is the infimum for the original problem actually aminimum?


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