Introduction to the Mathematical Theory of Control, Lecture 1

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Introduction to the Mathematical Theory of Control, Lecture 1

Monica Motta

Dipartimento di Matematica Università di Padova

Valona, September 11, 2017

Table of contents

1 Introduction

2 Examples

3 Topics addressed in this course

4 Equivalence Filippov’s Theorem

Introduction

Since the beginnings of Calculus, differential equations have provided an effective mathematical model for a wide variety of physical phenomena.

Consider a ’state’ x = (x₁, . . . ,x_n)and assume that the evolution t 7!^x(t) can be described by the ODE

x˙(t) =g(t,x(t)). (1) Under suitable assumptions on g, if the state of the system is known at some initial time t0:

x(t₀) =x₀, (2)

then the future (and past) behavior of the state x can be uniquely determined (Cauchy Problem). Celestial mechanics provides a typical example of this situation, where we can only predict but not modify the evolution of the system.

Control Theoryprovides a different paradigm. We now assume the presence of an external agent, i.e. a "controller", who can actively influence the evolution of the system. This new situation is modeled by a control system, namely

x˙(t) =f(t,x(t),u) u(·) 2 U, (3) whereU is a family of admissible control functions. In this case, the rate of changex˙(t)depends not only on the state x itself, but also on some external parameters, say u= (u1, . . . ,um), which can also vary in time.

The control function u(·), subject to some constraints, will be chosen by a controller in order to achieve certain preassigned goals:

steer the system from one state to another (controllability),

keep the state close to a given position in long time (stabilizability), minimize a certain cost functional (optimization), etc...

Open loop and closed loop controls

We may have

Open loop controls: u=u(t),

u(·) 2 U := {^u: [t₀,t₁]!^U =U ⇢ R^m, measurable}.

Given such u(·)^{, t} 7!^f(t,x(t),u(t))⌘^g(t,x(t))is a classical differential equation.

Closed loop or feedback controls: u=u(x)2^U.

Usually, x 7!^u(x)that performs a given task is discontinuous, hence t 7!^f(t,x(t),u(x))⌘^g(t,x(t))is NOT a classical differential

inclusion (discontinuous in x!).

Finding open loops controls is often easier, but feedback controls have the advantage that they do not need to be redetermined if the initial configuration is changed. Moreover, they are robust in the presence of random perturbations.

We will consider only open loop controls.

Differential inclusions

The control system can then be written as adifferential inclusion, namely x˙(t)2^F(t,x(t)) (4) where the set of possible velocities at any time t is given by

F(t,x) :={^f(t,x,u) : u2^U}

Clearly, every admissible trajectory of the control system (3) is also a solution of (4). Under some regularity assumptions on f , it turns out that the converse is also true.

Examples

Born in the ’60s, Control Theory has a lot of applications in MECHANICS, ENGINEERING, ECONOMICS, BIOLOGY, . . .

Example 1 (Boat on a river).

Consider a river with straight course. Using a set of planar coordinates, assume that it occupies the horizontal strip

S := {(^x1,x2) : x1 2 R, ^x2 2 [ ¹,1]} Assume that speed of the water is given by the velocity vector

v(x₁,x₂) = (1 x₂²,0). If the boat is powered by an engine, its motion can be modeled by the control system

( ˙x₁, ˙x₂) =v+u= (1 x₂²+u₁,u₂) (5) where the vector u describes the velocity of the boat relative to the water.

Let the control set U be the closed ball inR² of radius M>0.

Given an initial condition(x₁,x₂)(0) = (¯x₁, ¯x₂), solving (5) one finds 8<

:

x1(t) = ¯x1+t+Rt

0u1(s)ds Rt

0 x¯2+Rs

0 u2(s⁰)ds^{0 2} ds, x2(t) = ¯x2+R_t

0u2(s)ds

In particular, u⌘ ( ²/3,1)takes the boat from a point(¯x1, 1)on one side of the river to(¯x1, 1)on the opposite side in t =2.

Actually, the boat can be steered from any point to any other point.

Equivalent differential inclusion:

( ˙x1, ˙x2)2^F(x1,x2) :=

⇢

(y1,y2) : q

(y1 1+x₂²)²+y₂²^{M .}

Example 2 (Car parking).

Consider a car in a large parking lot. At a given time, its position is determined by the coordinates(x,y)of its barycenter B and the angle✓ giving its orientation. The driver controls the car by acting on the gas pedal and on the steering wheel. The controls are thus the speed u(t)of the car and the turning angle↵(t). The motion is described by the nonlinear control system

( ˙x, ˙y, ˙✓) = (u cos✓,u sin✓, ↵u). (6) It is natural to assume(u, ↵)2^U := [ m,M]⇥ [ ¯↵, ¯↵]. The typical maneuver needed for parallel parking is illustrated in the figure.

Call t 7!^x(t,u)the solution of the Cauchy problem

x˙(t) =f(t,x(t),u(t)) u(·) 2 U, ^x(0) = ¯x. (7) A wide range of mathematical questions can be formulated. We will address just some of them:

Concerned with the dynamics of the system:

Fundamental properties of trajectories and of the set of solutions Reachable set at time t: R(^t) :={^x(t,u) : u(·) 2 U}

Concerned with optimal control: among all strategies which accomplish a certain task, one seeks an optimal one, which minimizes/maximizes a given performance criterion, as, for instance, J = (T,x(T)).

Existence of optimal controls

Necessary conditions for the optimality of a control

Some (incomplete!) bibliography

AC J.P. Aubin & A. Cellina, Differential inclusions. Set-valued maps and viability theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264.

Springer-Verlag, Berlin, 1984.

BP A. Bressan, B. Piccoli, Introduction to the mathematical theory of control. AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.

RW R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften, 317, Springer-Verlag, New York, 1998.

V R. B. Vinter, Optimal Control. Birkhäuser, Boston, 2000.

an the references therein....

Properties of the control system

Let us now begin a study of the control system:

x˙(t) =f(t,x(t),u(t)) u(·) 2 U, ⁽⁸⁾ where

U := {^u(·) measurable, u(t)2U for all t} We shall assume:

(H) The control set U ⇢ R^m is compact,⌦is an open subset ofR ⇥ R^{n, f} is continuous in all the variables and C¹(continuously differentiable) in x.

Definition 3.

An absolutely continuous function x(·) : [^a,b]! R^n(x2^AC([a,b],Rⁿ)) is a solution to (8) if

graph x(·) := {(^t,x(t)) : t 2 [^a,b]} ⇢ ⌦^; x˙(t) =f(t,x(t),u(t))for a.e. t 2 [^a,b].

1. Equivalence between the controls system and a differential inclusion

Let us introduce the multi-function F : ⌦! P(Rⁿ)given by

F(t,x) :={^f(t,x,u) : u2^U}and consider the differential inclusion x˙(t)2^F(t,x(t)) a.e. t (9)

Theorem 4 (Equivalence Filippov’s Thm.).

Under assumption (H), a function x 2^AC([a,b],Rⁿ)is a solution to

x˙(t) =f(t,x(t),u(t)) (10) if and only if it verifies (9) for a.e. t 2 [^a,b].

Proof. It is based on the use of a Measurable Selection Thm. for multifunctions.

Basic facts on differential inclusions

Let X, Y be metric spaces. For any x 2^{X and A}⇢^{X, set} d(x,A) := inf

a2Ad(x,a); B(A, ") :={^x 2^X : d(x,A) < "}.

For any pair of nonempty, compact subsets A, A⁰⇢^{X, theirHausdorff} distanceis defined as

dH(A,A⁰) :=max{^d(x,A⁰), d(x⁰,A) : x2^A, x⁰ 2^A0}.

Equivalently,

dH(A,A⁰) :=inf{⇢ >⁰: A⇢^B(A⁰, ⇢), and A⁰ ⇢^B(A, ⇢)}.

d_H(A,A⁰) =max{⇢, ⇢⁰}

A multifunction F :X ! P(^Y)has compact values if F(x)⇢^{Y is} compact. It isHausdorff-continuousif for every x 2^X,

xlim⁰!xdH(F(x⁰),F(x)) =0

We say that the multifunction F hasclosed graphif its graph graph(F) :={(^x,y) : y 2^F(x)}is a closed subset of X⇥^{Y .} This condition means that,

whenever x⌫ !^{x, y}⌫ !^{y and y}⌫ 2^F(x⌫)for every⌫, then we also have the inclusion y 2^F(x).

Given a multifunction t 7!^F(t)with non-empty values, a natural problem is to construct aselection, i.e. a single-valued, function f such that

f(t)2^F(t)for every t with some additional properties, such as continuity or at least measurability.

To this aim, we introduce the notion oflexicographical orderingonR^n. Given x, y 2 R^{n, x} 6=y, we write x y if one of the following alternatives holds:

x1<y1

x₁=y₁, x₂ <y₂ . . .

x1=y1, x2=y2, . . . ,xn 1=yn 1, xn<yn. Notice the analogy with the ordering of words in a dictionary.

Any compact set K ⇢ Rⁿ has one first point⇠, w.r.t. the lexicographical order.

Examples of first points of a compact set K w.r.t. the lexicographical order.

Proof.

If t 7!^F(t)⇢ Rⁿis a multifunction with compact values, we can now define thelexicographic selectiont 7! ⇠(^t)2^F(t), where⇠(t)is the first point of the compact set F(t)w.r.t. the lexicographical order.

Theorem 5 (Measurable selection).

Let t 7!^F(t)be a bounded multifunction with closed graph, defined for t 2 [^a,b]. For each t, let⇠(t)2^F(t)be the lexicographic selection.

Then the map t 7! ⇠(^t)is measurable.

Proof.

Theorem 6 (Lusin’s Theorem).

For any f : [a,b]! Rⁿ, f is measurable if and only if there exists a sequence of disjoint, compact subsets Jk ⇢ [^a,b]with

meas [a,b]\ [⁺¹_k=1^Jk =0

and such that the restriction of f to each set Jk is continuous.

Global existence and continuous dependence

Under assumption (H), for any measurable control u(·) 2 U ^{and any} (t0, ¯x)2 ⌦, the Cauchy problem

x˙(t) =f(t,x(t),u(t)) =:g(t,x(t)), x(t₀) = ¯x (11) admits a unique, local solution t 7!^x(t,u)on[t₀ ,t₀+ ]for some

>0, since g verifies the so-called Carathéodory conditions (i.e., it is measurable in t and C¹ in x... see e.g. [BP]).

To guarantee the existence of a global, bounded solution, let us assume

(H)^⇤ The control set U ⇢ R^mis compact, f is continuous onR ⇥ Rⁿ⇥ U, C¹in x, and such that, for some constants C and L,

|f (t, x, u)|  C, kDxf (t, x, u)k  L for all (t, x, u).

Theorem 7 (Global existence and continuous dependence).

Under assumption (H)^⇤, for every T >0,

i) for any u(·) 2 U the Cauchy problem

x˙(t) =f(t,x(t),u(t)), x(t0) = ¯x (12) has a unique solution x(·,^u)defined for all t 2 [⁰,T].

ii) the input-output map u(·) 7!^x(·,^u(·))is continuous from L¹([0,T],R^m)into C⁰([0,T],Rⁿ).

Proof. The main ingredients of the proof are:

the Contraction Mapping Theorem and the Dominated Convergence Theorem recalled below.

Theorem 8 (Contraction Mapping).

Let X be a Banach space,⇤a metric space, and let : ⇤⇥^X !^{X be a} continuous mapping such that, for some` <1,

k ( ,^x) ( ,y)k  `k^{x y}k.

Then for each 2 ⇤there exists a fixed point x( )2X such that x( ) = ( ,x( )).

The map 7!^x( )is continuous. Moreover, for any 2 ⇤^{, y} 2^{X one has}

k^{y x}( )k  ₁¹ `k^y ( ,y)k.

Proof.

Theorem 9 (Dominated Convergence).

Let v, , v1, . . . ,v⌫, . . . be functions in L¹([0,T],R^N)verifying v⌫(t)!^v(t)for a.e. t2 [⁰,T];

|^v⌫(t)|  (^t)for all⌫ 1 and a.e. t2 [⁰,T]. Then

⌫!+1lim Z _T

0 v_⌫(t)dt = Z _T

0 v(t)dt. Moreover

If v, v₁, . . . ,v⌫, . . . are functions in L¹([0,T],R^N)such that v⌫ !^{v in L1,} then one can extract a subsequence(v_⌫0)⌫⁰ such that

v_⌫0(t)!^v(t) for a.e. t 2 [⁰,T].

Lecture 1

mercoledì 6 settembre 2017 19:20

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