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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

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 2 / 32

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)

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 3 / 32

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.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 5 / 32

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!)

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 6 / 32

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?

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 7 / 32

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?

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 7 / 32

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⌘.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 8 / 32

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)

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 9 / 32

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

).

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 10 / 32

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).

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 11 / 32

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.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 12 / 32

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.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 13 / 32

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.

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

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.

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

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).

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 15 / 32

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).

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 16 / 32

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.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 17 / 32

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.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 18 / 32

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.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 19 / 32

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))

with initial and terminal constraints

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

has an optimal solution.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 20 / 32

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))

with initial and terminal constraints

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

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 21 / 32

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.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 22 / 32

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.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 23 / 32

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.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 24 / 32

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.

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 25 / 32

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?

M. Motta (Padua Un.) Control Theory Valona, September 14, 2017 26 / 32

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