Introduction to the Mathematical Theory of Control, Lecture 4

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Introduction to the Mathematical Theory of Control, Lecture 4

Monica Motta

Dipartimento di Matematica Università di Padova

Valona, September 14, 2017

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⁾ Thereachable setR(⌧, ¯^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 a suitable convexity assumption. This closure property will be of great importance, 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) : u2^U}are convex then, for every⌧ 2 [⁰,T], the reachable setR(⌧, ¯^x)is compact.

Remark. More generally, an analogous result holds true for the reachable set at time⌧, starting from points in some set K⁰, that is

R(⌧,^K0) :={^x(⌧ ) : x(·)solution of (1) for somex¯2^K0}.

Moreover, everything can be extended to t-dependent data.

Example 2.

Consider the system onR²

( ˙x₁, ˙x₂) = (u,x₁²) (x₁,x₂)(0) =0, u(t)2^U :={ ¹,1} Consider the sequence of rapidly oscillating controls of the previous examples. Then, given T >0, the corresponding trajectories converge uniformly to(x1,x2)(·) ⌘ (⁰,0), but this is NOT a trajectory of the system.

Indeed, if(x1,x2)(·)is a solution, thenx˙1(t)2 { ¹,1}^{implies x}1(t)6=^{0 at} a.e. time t. Hence theR(^T,0)is not closed, since

x2(T) =RT

0 x₁²(t)dt >0.

Example 3.

Consider the scalar system

x˙ =u x² x(0) =1, u(t)2^U := [ 1,1].

Notice that F(x) = [ 1,1]x²has compact and convex values (linear system in u!). For each⌫ 1, define the control

u⌫(t) =

( _{1 t}

2 [⁰,1 (1/⌫)]

0 t>1 (1⌫) =)^x⌫(t) =

( ₁

1 t t 2 [⁰,1 (1/⌫)]

⌫ t >1 (1/⌫) On every[0, ⌧ ]with⌧ 1, there is not a uniform bound on this set of solutions. In particular, x⌫(⌧ ) = ⌫, so that the reachable set is not bounded.

( As a consequence, the optimization problem of maximizing x(T)for a fixed T 1 has supremum+1and 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]toR^n.

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 for applications?

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]toR^n.

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 for applications?

A priori bounds

The next theorem establishes asufficient condition for the validity of hypothesis P.1in 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)| ^eCt|¯^x| +⇣

e^Ct 1⌘ .

Chattering controls

In many practical situations hypothesis P.2 does not hold, namely the sets of admissible velocities F(x) ={^f(x,u) : u2^U}are not convex, hence the reachable sets may not be closed.

In this case one can provide a representation of the closure of the reachable 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 ⇢ R^n,

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 be

represented as a convex combination of at most n+1 elements in K :

co K = ( _n

X

i=0

✓_ik_i : (✓0, . . . , ✓_n)2 rn, k_i2K for all i )

,

where

rn :=

(

✓ = (✓₀, . . . , ✓_n) : Xn

i=0

✓_i =1, ✓_i 0 for all i )

.

As a consequence,

F^](x) =co F(x) = ( _n

X

i=0

✓_if(x,ui) : (✓0, . . . , ✓_n)2 rn, ui 2U 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)) :=

Xn

i=0

✓_if(x,ui).

Generalized controls of the form u^]are calledchattering controls.

In practical applications, they can be approximated by rapidly switching the control value u(t) among the values u₀(t), . . . un(t), with the length of time during which u = u_iproportional 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 closureR(⌧, ¯^x)of the reachable set for the system (5) coincides with the (compact) reachable setR^](⌧, ¯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 U⁰ ⇢U such that

co{^f(x,u) : u⁰2^U} ◆ {^f(t,x,u) : u2^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^U0 uniformly on bounded intervals.

Example 8.

Consider again the system onRintroduced yesterday:

x˙(t) =u(t), x(0) =0, u(t)2^U0 ={ ¹,1}.

Observe that U⁰ ⇢^U :=co(U⁰) = [ 1,1]and

co{^f(t,x,u) : u2^U0} = {^f(t,x,u) : u2^U} The approximation Thm. says that any trajectory corresponding to u(t)2 [ ¹,1]can be approximated by trajectories with controls taking only 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 the chattering system can be approximated by trajectories of the original system.

Example 8.

Consider again the system onRintroduced yesterday:

x˙(t) =u(t), x(0) =0, u(t)2^U0 ={ ¹,1}.

Observe that U⁰ ⇢^U :=co(U⁰) = [ 1,1]and

co{^f(t,x,u) : u2^U0} = {^f(t,x,u) : u2^U} The approximation Thm. says that any trajectory corresponding to u(t)2 [ ¹,1]can be approximated by trajectories with controls taking only 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 the chattering system can be approximated by trajectories of the original system.

Let x⌫(·)be the trajectories associated to the controls (k integers) u⌫(t) =1 if k⇡/⌫ ^t  (^k+1)⇡/⌫; u⌫(t) = 1 otherwise.

The trajectories x_⌫(·)^converge uniformly to x(·) ⌘^{0 on}R^.

We can now observe that x(·) ⌘0 is NOT a solution for the original control system, 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) : u2^U}are not convex, then we have seen that the reachable sets may not be closed. A noteworthy exception occurs in the case ofsystems 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 also by trajectories of the original system

Theorem 9 (Reachable sets for linear systems).

Assume that U ⇢ R^m 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 setR(⌧, ¯^x)for system (9) is a compact, convex subset ofRⁿ. 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 f⁰, . . . ,f^k 2^L1([a,b],Rⁿ)be integrable vector valued functions. Let

✓₀, . . . , ✓_k : [a,b]! [⁰,1]be measurable weight functions such that P_k

i=0✓_i(t) =1 for every t.

Then there exist a partition of[a,b]into disjoint measurable subsets J0, . . . ,Jk such that

Z b

a

Xk

i=0

✓i(t)fⁱ(t)

! dt=

Xk

i=0

Z

Ji

fⁱ(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 the admissible controls take values in aconvex polytopewith vertices w₁, . . . ,w_N 2 R^m

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 the polytope. In this case, the admissible control functions u(·)^{are called} bang-bang controls. Indeed, they must be piecewise constant, jumping between 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 and coincide.

Mayer problem with T fixed

Consider the usual control system

x˙(t) =f(t,x(t),u(t)) u2 U. ⁽⁸⁾ Given T >0, an initial state¯x, a set of admissible terminal conditions S ⇢ Rⁿ, and a cost function :Rⁿ ! Rwe consider the optimization problem

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 ! Rbe continuous,S ⇢ Rⁿclosed, 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) : u2^U}are convex.

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

minu2U (x(T,u)) with initial and terminal constraints

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)2U for every t} Given an initial statex, a set of admissible terminal conditions¯ S ⇢ R ⇥ Rⁿ, and a cost function :R ⇥ Rⁿ! Rwe consider the optimization problem

T >0, u2Umin (T,x(T,u)) with initial and terminal constraints

x(0) = ¯x, (T,x(T))2 S.

We shall assume

(Hg) The control set U ⇢ R^m is compact, f : [0, +1) ⇥ Rⁿ⇥^{U is}

continuous in all the variables , C¹ 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]⇥ Rⁿclosed, and moreover P.2 all sets of velocities F(t,x) :={^f(t,x,u) : u2^U}are convex.

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

The proof is a typical example of theDirect Methodfor proving the existence 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 the appropriate initial and terminal conditions.

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

Extensions

Themaximization problem

T >0, u2Umax (T,x(T,u)) is of course equivalent choosing = . Theminimization problem of Bolza

minu2U

⇢Z T

0 L(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

x˙0 =L(t,x(t,u),u(t)), x0(0) =0.

Convexity assumption

In the case oflinear 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]⇥ Rⁿclosed and U compact.

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

Proof.

Moreover, if the control set is apolytope, one can choose the optimal control to bebang-bang.

In the general non-convex case, one can often prove the existence of an optimal chattering control, for the generalized optimization problem where F(t,x) ={^f(t,x,u) : u2^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 a minimum?