An introduction to Mathematical Theory of Control

An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 1 / 120

Introduction Since the beginnings of Calculus, differential equations have provided an effective mathematical model for a wide variety of physical phenomena. Consider a system whose state can be described by a finite number of real valued parameters, say x = (x 1, x 2,..., x n ). If the rate of change ẋ = dx dt is entirely determined by the state x itself, then the evolution of the system can be modelled by the ordinary differential equation ẋ = g (x). (1) If the state of the system is known at some initial time t 0, the future behavior for t > t 0 can then be determined by solving a problem of Cauchy consisting of (1) together with the initial condition x (t 0 ) = x 0. (2) Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 2 / 120

Introduction We are here taking a spectator s point of view: the mathematical model allows us to understand a portion of the physical word and predict its future evolution, but we have no means of altering its behavior in any way. Celestial mechanics provides a typical example of this situation. We can accurately calculate the orbits of moons and planets and exactly predict time and location of eclipses, but we cannot change them in the slightest amount. This classical point of view assumes that physical systems are isolated and all the information contained in them is completely accessible to the external observer. The mathematical model describes the evolution of the system taking into account its internal strengths and observations on the systems made from abroad but there is no way to interfere in this evolution. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 3 / 120

Introduction The failure of this framework turns out to be clear in front of the development of technology and engineering. The invention of machines and tools and their dissemination put the researcher in front of the reality of systems that can be easily guided from abroad in order to regulate its operation based on a pre-established project and with the specific objective to be realized. Mathematical models of these systems should take into account also the forces or impulses coming from abroad. Thus, comes the need to accept the modern point of view, where any physical system (machine, or chemical, biological or social process) is defined in a certain region of space and has its own dynamics in base which develops, exchanging continuously information with the external environment. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 4 / 120

Introduction Such exchange of information occurs through certain channels that are named generically inputs (if the information coming from the exterior to the system) and outputs (the information will be to the outside of the system). Graphically we can represent a system like u (t) x (t) The mathematical control theory is the theoretical basis for the study of physical systems equipped with inputs and outputs and provides a different paradigm. We now assume the presence of an external agent, namely a "controller" who can actively influence the evolution of the system. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 5 / 120

Introduction This new situation is modelled by a control system, ẋ = f (x, u), u (.) U (3) where U is a family of admissible control functions. In this case, the rate of change ẋ (t) depends not only of the state x itself, but also on some external parameters, say u = (u 1, u 2,..., u m ), which can also vary in time. The control function u (.) subject to some constraints, will be chosen by a controller in order to modify the evolution of the system and achieve certain preassigned goals: steer the system from a state to another, maximize the terminal value of one of the parameters, minimize a certain cost functional, etc. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 6 / 120

Introduction In a standard setting, we are given a set of control values U R m. The family of admissible control functions is defined as U := {u : R R m : u measurable, u (t) U for a. e. t} The control system (3) can be written as a differential inclusion, namely x F (x) (4) where, for each x, the set of possible velocities is given by F (x) := {y : y = f (x, u) for some u U}. (5) Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 7 / 120

Introduction An absolutely continuous function x : [0, T ] R n is said to be an admissible solution (or trajectory) of the control system (3) if there exists a measurable control u : [0, T ] U such that x (t) = f (x (t), u (t)) for almost every t [0, T ]. (6) By solution (or trajectory) of the differential inclusion (4) we mean any absolutely continuous function x : [0, T ] R n such that x (t) F (x (t)) for almost every t [0, T ]. (7) Clearly, every admissible trajectory of the control system (3) is also a solution of the differential inclusion (4). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 8 / 120

Introduction Under some regularity assumptions on f, given any solution of (4), one can select a measurable control u : [0, T ] U such that x (t) = f (x (t), u (t)) for almost every t [0, T ], that is, x is a trajectory of the control system (3). Differential inclusions often provide a convenient approach for the analysis of control systems, as we will see in what follows. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 9 / 120

Introduction The control law can be assigned in two basically different ways: in "open loop" form, as a function of time: t u (t), and in "closed loop" or "feedback", as a function of the state: x u (x). Implementing an open loop control u = u (t) is in a sense easier, since the only information needed is provided by a clock, measuring time. On the other hand, to implement a closed loop control u = u (x) one constantly needs to measure the state x of the system. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 10 / 120

Introduction Designing a feedback control, however yields to some distinct advantages. The feedback controls are more robust in the presence of random perturbations. For example, assume that we seek a control u (.) which steers the system described by (3) from an initial state P to the origin. This can be achieved say by an open loop control t u (t). In many practical situations, however, the evolution is influenced by additional disturbance which cannot be predicted in advance. The actual behavior of the system will be thus be governed by ẋ = f (x, u) + η (t), (8) where t η (t) is a perturbation term. In this case, if the open loop control u = u (t) steers the system (3) from an initial state P to the origin, this same control function may not accomplish the same task with (8), when a perturbation is present. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 11 / 120

Introduction Alternatively, one can solve the problem of steering the system described by (3) from an initial state P to the origin by mean of a closed loop control. In this case, we would seek a control function u = u (x) such that all the trajectories of the ordinary differential equation ẋ = g (x) := f (x, u (x)) (9) approach the origin as t. This approach is less sensitive to the presence of external disturbances. Two examples will be presented below, Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 12 / 120

Example 1: boat on the river Consider a river with a straight course, and using a set of planar coordinates, assume that it occupies the horizontal strip S := { (x 1, x 2 ) R 2 : < x 1 < +, 1 < x 2 < 1 }. Moreover, assume that speed of the water is given by the velocity vector v (x 1, x 2 ) = ( 1 x 2 2, 0 ). If a boat on the river is merely dragged by the current, its position will be determined by the differential equation (ẋ 1, ẋ 2 ) = ( 1 x 2 2, 0 ). On the other hand, if the boat is powered by an engine, then its motion can be modelled by the control system (ẋ 1, ẋ 2 ) = ( 1 x 2 2 + u 1, u 2 ), (10) where u = (u 1, u 2 ) describe the velocity of the boat relative to the water. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 13 / 120

Example 1 The set U of admissible controls consists of all measurable functions u : R R 2 taking values inside the closed disc { } U = (ω 1, ω 2 ) R 2 : ω 21 + ω22 M, (11) where M accounts for the maximum speed (in any direction) that can be produced by the engine. Given an initial condition (x 1, x 2 ) (0) = ( x 1, x 2 ) solving (10) one finds t t x 1 (t) = x 1 + t + u 1 (s) ds x 2 + 0 t x 2 (t) = x 2 + u 2 (s) ds ( 1 x 2 1). 0 0 s 0 u 2 (τ) dτ 2 ds, Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 14 / 120

Example 1 In particular, the constant control u = (u 1, u 2 ) = ( 2 3, 1) takes the boat from a point ( x 1, 1) on one side of the river to the point ( x 1, 1) on the opposite side, in two units of time. Observe that if M > 0 then the boat can be steered from any point P of the river to any other point of the river (exercise: justify!) and that the admissible trajectories of the control system (10) (11) coincide with the solutions of the differential inclusion (ẋ 1, ẋ 2 ) F (x 1, x 2 ) = { (y 1, y 2 ) : (y 1 1 + x 22 )2 + y 22 M }. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 15 / 120

Example 2: cart on a rail Consider a cart which can move without friction along a straight rail. For simplicity assume that it has unit mass. Let y (0) = ȳ be its initial position and v (0) = v be its initial velocity. If no forces are present, its future position is simply given by y (t) = ȳ + t v. Next, assume that a controller is able to push the cart, with an external force u = u (t). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 16 / 120

Example 2 The evolution of the system is then determined by the second order equation (determined by the Newton law) ÿ = u (t). (12) Calling now x 1 (t) = y (t) and x 2 (t) = v (t), respectively the position and the velocity of the chart at time t, the equation (12) can be written as a first order control system { ẋ1 = x 2 (13) ẋ 2 = u Given the initial condition x 1 (0) = ȳ, x 2 (0) = v, solving (13) one finds t x 1 (t) = ȳ + t v + 0 t (t s) u (s) ds, x 2 (t) = v + 0 u (s) ds. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 17 / 120

Example 2 Assuming that the force satisfies the constraint u (t) 1, the control system (13) is equivalent to the differential inclusion (ẋ 1, ẋ 2 ) F (x 1, x 2 ) = {(x 2, ω) : 1 ω 1}. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 18 / 120

Example 2 We now consider the problem of steering the system at the origin (0, 0) R 2. In other words, we want the chart to be eventually at the origin with zero speed. For example, if the initial condition (ȳ, v) = (2, 2), this goal is achieved by the open-loop control 1 if 0 t 4 ū (t) = 1 if 4 t 6 0 if t 6. A direct calculation shows that (x 1 (t), x 2 (t)) = (0, 0) for t 6. However, the above control would not accomplish the same task in connection with (ȳ, v) = (2, 2) (exercise: verify!) Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 19 / 120

Example 2 A related problem is that of asymptotic stabilization: we seek a feedback control function u = u (x 1, x 2 ) such that for every initial data (ȳ, v) the corresponding solution of the Cauchy problem (ẋ 1, ẋ 2 ) = (x 2, u (x 1, x 2 )), (x 1, x 2 ) (0) = (ȳ, v) approaches the origin as t, that is, lim (x 1, x 2 ) (t) = (0, 0). t There are several feedback controls which accomplish this task. One is u (x 1, x 2 ) = x 1 x 2. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 20 / 120

Questions of interest In connection with a control system of the general form (3) several mathematical questions can be formulated. A first set of problems is concerned with the dynamics of the system. Given an initial state x, one would like to determine which other states x R n can be reached using the various admissible controls u U. More precisely, given a control function u = u (t), call x (., u) the solution of the Cauchy problem ẋ (t) = f (x (t), u (t)), x (0) = x, and define the reachable set at time t as R (t) = {x (t, u) : u U}. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 21 / 120

Questions of interest The closure, boundedness, convexity and the dimension of the set R (t) provide useful information on the control system. In addition, it is interesting to study whether R (t) is a neighborhood of the initial point x for all t > 0. In the positive case, the system is said to be small time locally controllable at x. Another important case is when R (t) includes the entire space t>0 R n. We then say that the system is globally controllable. The dependence of the reachable set R (t) on the time t and on the set of admissible controls U is also a subject of investigation. One may ask whether the same points in R (t) can be reached by using controls which are piecewise constant, or take values within the set of extreme points of U. Being able to perform the same tasks by mean of a smaller set of control functions, easier to implement, is quite relevant in practical applications. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 22 / 120

Questions of interest A second very important area of control theory is concerned with the optimal control. In many applications, among all strategies which accomplish a certain task, one seeks an optimal one, based on a given performance criterion. A performance criterion can be defined by an integral functional of the form J = T 0 L (t, x, u) dt, (14) and the values of J will have to be optimized among the admissible trajectories of (3), with a number of initial or terminal constraints. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 23 / 120

Questions of interest For example, among all controls which steer the system from the initial point x to some point on the target set T at time T, we may seek the one that minimize the cost functional (14). This problem is formulated as T min u U 0 L (t, x, u) dt, (15) subject to ẋ = f (x, u), x (0) = x, x (T ) T. (16) Observe that if (3) takes the simplest form ẋ = u, u (t) U = R n and if T = {ȳ} consists on just one point, then we do not have any constraint on the derivative ẋ. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 24 / 120

Questions of interest Our problem of optimal control thus reduces to the standard problem of the Calculus of Variations min x (.) T 0 L (t, x, ẋ) dt, x (0) = x, x (T ) = ȳ. (17) Roughly speaking, the main difference between the problem (15) (16) and (17) is that in (17) the derivative ẋ is not restricted, while in (15) (16) the derivative is constrained within the set F (x) := {y : y = f (x, u) for some u U}. The basic mathematical theory of optimal control has been concerned with the following main topics: existence of optimal controls; necessary conditions for the optimality of a control; suffi cient conditions for optimality. The next section will be dedicated to Carathéodory solutions to ordinary differential equations which arise from control systems, and to which the classical notion of solution is not appropriate. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 25 / 120

Normed vector spaces A real vector space is a nonempty set X endowed with two operations: + : X X X and : R X X satisfying the following conditions: 1 (x + y) + z = x + (y + z), x, y, z X ; 2 x + y = y + x, x, y X ; 3 there exists 0 X such that x + 0 = x, x X ; 4 for every x X there exists x X such that x + ( x) = 0 x X ; 5 (α + β) x = α x + β x, α, β R, x X ; 6 α (x + y) = α x + α y, α R, x, y X ; 7 1 x = x, x X (where 1 is the unit real number) 8 α (β x) = (αβ) x, α, β R, x X. We denote by (X, +, ) a vector space with the operations + and. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 26 / 120

Normed vector spaces A subset A of a vector space (X, +, ) is said to be convex if λx + (1 λ) y A,, x, y A and λ [0, 1]. Let (X, +, ) be a (real) vector space. A norm on X is a mapping. : X R with the following properties: (i) x = 0 if and only if x = 0; (ii) α x = α x α R, x X ; (iii) x + y x + y x, y X. If. is a norm on X then we say that (X,. ) is a vector normed space, or simply a normed space. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 27 / 120

Normed vector spaces Let (X,. ) be a normed space and for x 0 X and r > 0 let B r (x 0 ) = {x X : x x 0 < r} be the open ball centered at x 0 with radius r, and B r (x 0 ) = {x X : x x 0 r} be the closed ball centered at x 0 with radius r. Both the open and the closed balls centered at x 0 with radius r are convex set Indeed, if x, y B r (x 0 ) and 0 λ 1 then λx + (1 λ) y x 0 λ (x x 0 ) + (1 λ) (y x 0 ) = λ x x 0 + (1 λ) y x 0 < λr + (1 λ) r = r hence λx + (1 λ) y B r (x 0 ), and B r (x 0 ) is a convex set. The proof for B r (x 0 ) is similar. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 28 / 120

Normed vector spaces are metric spaces Recall that if (X,. ) is a normed space then d : X X R defined by d (x, y) = x y is a distance (or metric) on X, that is satisfies the following properties: (a) d (x, y) = 0 if and only if x = y; (b) d (x, y) = d (y, x) x, y X ; (c) d (x, y) d (x, z) + d (z, y) x, y, z X. If d : X X R satisfies (a), (b) and (c) then d is called a metric on X and (X, d) is said to be a metric space. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 29 / 120

Banach spaces Let (X,. ) be a normed space. A sequence (x n ) n 1 X is said to be convergent if there exists x X such that: for every ε > 0 there exists n ε N so that x n x < ε whenever n n ε. A sequence (x n ) n 1 X is said to a Cauchy sequence if for every ε > 0 there exists n ε N so that x n x m < ε whenever n, m n ε. We say that a normed space is complete if every Cauchy sequence is convergent. A complete normed vector space is called a Banach space. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 30 / 120

Examples of Banach spaces The finite dimensional space R n with the Euclidian norm x = n i=1 x i, if x = (x 1,..., x n ) R n ; The space C ([a, b], R n ) of all continuous functions f : [a, b] R n, with norm f = sup f (t) ; t [a,b] Given a subset Ω R m, we say that a map f : Ω R n is Lipschitz continuous if there exists a constant L such that f (x) f (y) L x y, x, y Ω. The space Lip (Ω, R n ) of all these Lipschitz continuous mappings is a Banach space with norm f Lip = sup f (x) + sup x Ω x =y f (x) f (y). x y Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 31 / 120

Examples of normed space which is not Banach spaces The space X = P ([0, 1]) of polynomial functions p : [0, 1] R with the norm p = sup p (t) t [0,1] is a normed space but not a Banach space Indeed, the sequence of polynomials p n (t) = n k=0 t k k! is a Cauchy sequence converges uniformly on [0, 1] hence also with respect to. to the continuous function f (t) = e t. Hence it is a Cauchy sequence in (C ([0, 1], R),. ), therefore also in (P ([0, 1]),. ). But its limit f (t) = e t, is not polynomial, that is,it has no limit in the space X = P ([0, 1]). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 32 / 120

Banach contraction mapping principle The next Banach s theorem show that for an equation, a unique solution exists and depends continuously on the parameters that describe the problem if the equation can be written in the form x = Φ (λ, x) (18) and the map x Φ (λ, x) is a contraction for each given value of the parameter λ, that is, for some κ < 1, Φ (λ, x) Φ (λ, y) κ x y, λ Λ, x, y X. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 33 / 120

Banach contraction mapping principle Theorem (contraction mapping principle) Let (X,. ) be a Banach space, Λ be a metric space and let Φ : Λ X X be a continuous mapping such that, for some κ < 1, Φ (λ, x) Φ (λ, y) κ x y, λ Λ, x, y X. (19) Then: (i) For each λ Λ there exists a unique x (λ) X such that (ii) The map λ x (λ) is continuous (iii) For any λ Λ, y X one has x (λ) = Φ (λ, x (λ)) (20) y x (λ) 1 y Φ (λ, y). (21) 1 κ Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 34 / 120

Proof of Banach contraction principle Fix any point y X. For each λ Λ, consider the sequence y k = y k (λ) defined by y 0 = y, y k+1 = Φ (λ, y k ) for k 0. By induction, for every k 0 one checks that y k+1 y k κ k y 1 y 0 = κ k y Φ (λ, y). (22) Since κ < 1, the sequence (y k ) k 0 is Cauchy in the Banach space X, hence it converges to some limit point, which we call x (λ). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 35 / 120

Proof of Banach contraction principle By the continuity of Φ (λ,.) we have hence (20) holds. x (λ) = lim k y k+1 = lim k Φ (λ, y k ) = Φ (λ, x (λ)), The uniqueness of x (λ) is proved observing that, if then by (19) it follows that x 1 = Φ (λ, x 1 ), x 2 = Φ (λ, x 2 ), x 1 x 2 = Φ (λ, x 1 ) Φ (λ, x 2 ) κ x 1 x 2. The assumption κ < 1 implies x 1 = x 2. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 36 / 120

Proof of Banach contraction principle Next, observe that (22) yields y k+1 y k i=0 y i+1 y i y Φ (λ, y) y Φ (λ, y) and letting k we obtain (21). i=0 κ i = k κ i i=0 y Φ (λ, y). 1 κ To show the continuity of λ x (λ) let (λ n ) n 1 be a sequence of parameters converging to λ. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 37 / 120

Proof of Banach contraction principle Using (21) with λ = λ n and y = x (λ ) we obtain x (λ ) x (λ n ) 1 1 κ x (λ ) Φ (λ n, x (λ )) = 1 1 κ Φ (λ, x (λ )) Φ (λ n, x (λ )) (23) and since λ Φ (λ, x) is continuous, the right-hand side of (23) tends to zero as n. Hence x (λ n ) x (λ ) proving the continuity of λ x (λ). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 38 / 120

Elements of Lebesgue measure theory If I = (a, b) is an open interval of R then the Lebesgue measure of I is defined by µ (I ) = b a. For any open set A R the Lebesgue measure is defined by { } µ (A) = sup µ (I j ) : I j A, I j open int., I j I k = if j = k. j J j J If F R is compact (that is, closed and bounded), then there exists an open and bounded interval I = (α, β) such that F I. Define then the Lebesgue measure of F by µ (F ) = µ (I ) µ (I \F ). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 39 / 120

Elements of Lebesgue measure theory If X R is any subset, then we define and µ (X ) = sup {µ (F ) : F compact X } µ (X ) = inf {µ (A) : A compact open, bounded and with X A}. We say that X is Lebesgue measurable if µ (X ) = µ (X ) and the common value is called the Lebesgue measure of X and denoted by µ (X ) : µ (X ) := µ (X ) = µ (X ). One has the following: Proposition: A subset X R is Lebesgue measurable if and only if for every ε > 0 there exists a compact K ε R and an open set A ε R such that K ε X A ε and µ (A ε \K ε ) < ε. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 40 / 120

Elements of Lebesgue measure theory Example: A finite or countable set A R is Lebesgue measurable and has a null measure. We denote by L the family of all Lebesgue measurable subsets of R and if I is an interval, we denote by L (I ) the family of all measurable subsets of I : L (I ) = {A L : A I }. Let I R be an interval. A function f : I R n is said to be Lebesgue measurable if for every open set V R n the preimage f 1 (V ) = {t I : f (t) V } is a Lebesgue measurable subset of I. Recall that f : I R n is said to be continuous if at each t 0 I, one has f (t 0 ) = lim t t0 f (t). f : I R n is said to be lower semicontinuous if at each t 0 I, one has f (t 0 ) lim inf t t 0 f (t). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 41 / 120

Elements of Lebesgue measure theory Remark: Every continuous or lower semicontinuous function, is measurable. We say that a property P holds almost everywhere (a.e. for short) on a set A R if there exists a set N R such that µ (N) = 0 and every point of A\N has the property P. Let (f n ) n 1 be a sequence of measurable functions, f n : I R n and if then the function f is measurable. f n (x) f (x) for a.e. x I If f : [a, b] R n is measurable and if f (t) φ (t) for all t [a, b] for some integrable scalar function φ, then f itself is integrable. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 42 / 120

Elements of Lebesgue measure theory If f and f differ only on a set of measure zero are identified. With this equivalence relation, the space of integrable functions f : [a, b] R n is written L 1 ([a, b], R n ) and it is a Banach space with norms f L 1 = b a f (t) dt. Lebesgue dominated convergence theorem: If (f n ) n 1 is a sequence of measurable functions which converges a.e. to f and if there exists an integrable function ψ such that f n (t) ψ (t) for all n 1 and t [a, b], then b a b f (t) dt = lim f n (t) dt n a Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 43 / 120

Elements of Lebesgue measure theory Characterization of measurable functions: For any function f [a, b] R n, the following statements are equivalent: (i) f is measurable (ii) For every ε > 0 there exists a compact set K ε [a, b] with and such that the restriction µ ([a, b] \K ε ) < ε f Kε is continuous. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 44 / 120

Elements of Lebesgue measure theory Absolutely continuous functions: A function f : [a, b] R n is said to be absolutely continuous if for every ε > 0 there exists δ > 0 such that for any finite family of disjoint intervals {]s i, t i ] : 1 i N} with total length N t i s i < δ one has i=1 N f (t i ) f (s i ) < ε. i=1 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 45 / 120

Elements of Lebesgue measure theory If f : [a, b] R n is absolutely continuous, then its derivative f is defined almost everywhere and it satisfies t f (t) = f (t 0 ) + f (s) ds for all t, t 0 [a, b]. t 0 Every Lipschitz continuous function f defined on [a, b] is absolutely continuous. The set of all absolutely continuous functions f : [a, b] R n is denoted by AC ([a, b], R n ). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 46 / 120

Elements of multivalued analysis Assume (X, d) is a metric space. For x X and A X let d (x, A) be the distance from x to A defined by d (x, A) = inf {d (x, y) : y A}. If A X and ε > 0 then B (A, ε) denotes the ε neighborhood of A defined by B (A, ε) = {x X : d (x, A) ε}. If A = {x 0 } then B (A, ε) is the closed ball centered at x 0 with radius ε : B (x 0, ε) = {x X : d (x, x 0 ) ε}. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 47 / 120

Elements of multivalued analysis If A, B are two closed, bounded and nonempty subsets of X then the Hausdorff-Pompeiu distance between A and B is defined by { } d H (A, B) = max sup d (a, B), sup d (b, A). a A b B One has that d H (A, B) = inf { ε > 0 : B B (A, ε) and A B (B, ε) } = max {d (a, B), d (b, A) : a A and b B}. The space C (X ) of all closed, bounded and nonempty subsets of X endowed with the Hausdorff-Pompeiu distance d H is a metric space. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 48 / 120

Elements of multivalued analysis Let X and Y be two sets and let 2 Y = {A : A Y } be the family of all subsets of Y. By multifunction or multivalued map or set-valued map from X to Y we mean a map or correspondence which associate to each x X a unique subset F (x) of Y. The set F (x) is called the value of F at x. A multifunction F from X to Y will be denoted by F : X 2 Y. The set Gr (F ) = {(x, y) : x X, y F (x)} is called the graph of the multifunction F : X 2 Y. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 49 / 120

Elements of multivalued analysis Let X, Y be metric spaces. We say that F is closed, bounded, compact valued if its values are all closed, bounded or compact. We say that F : X 2 Y has closed graph if Gr (F ) is a closed subset of the product metric space X Y, which is equivalent to: if: x n X, x n x, y n F (x n ), y n y then y F (x). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 50 / 120

Elements of multivalued analysis Let F : X 2 Y be a multivalued map with closed and bounded values. We say that F is Hausdorff continuous at x 0 X if lim d H (F (x), F (x 0 )) = 0, x x 0 that is, for every ε > 0 there exists δ ε > 0 such that d H (F (x), F (x 0 )) < ε whenever x B (x 0, δ ε ). Let I R and let F : I 2 Rn be a multifunction. A function f : I R n is said to be a selection of F if f (t) F (t) for all t I. If f : I R n is a selection of F and if it is a measurable function then we say that f is a measurable selection of F. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 51 / 120

Elements of multivalued analysis In order to introduce the lexicografic measurable selection, we need to define the lexicographic order relation on R n : if x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ) then we say that x < y if one of the following situations happens: x 1 < y 1 or x 1 = y 1 and x 2 < y 2 or... x 1 = y 1,...,x n 1 = y n 1, and x n < y n. One has the following: Proposition: If K R n is compact subset of R n then there there exists a first element ξ of K with respect to the lexicografic order, that is, there exists ξ K such that ξ < x for all x K. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 52 / 120

Elements of multivalued analysis If F : I 2 Rn is a multifunction with compact values then we can define the lexicographic selection t ξ (t) where ξ (t) F (t) is the first element of the compact set F (t) with respect to the lexicographic order. One has the following: Proposition If I = [a, b] and if F : I 2 Rn is a compact valued multifunction, with closed graph and if t ξ (t) is the lexicografic selection of F then t ξ (t) is a measurable selection. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 53 / 120

Carathéodory solutions As we already seen in the basic model of a control system a control system, ẋ = f (x, u) as soon as a control function u = u (t) from the family U of admissible controls is assigned, the evolution can be determined by solving the ordinary differential equation (ode, for short) ẋ = g (t, x) (24) where g (t, x) := f (x, u (t)). (25) Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 54 / 120

Classical solutions In the classical theory of ordinary differential equations like (24) we assume that the function g is continuous with respect to both the variable and the appropriate notion of solution is the one of "classical solution". Definition: Let Ω R R n be an open set and let g : Ω R n be a function which define the equation (24). A function ϕ : I R R n, with I an interval, is called a classical solution of (24) if ϕ C 1 (I, R n ), G (ϕ) := {(t, ϕ (t)) : t I } Ω and ϕ (t) = g (t, ϕ (t)) for all t I. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 55 / 120

Carathéodory solutions For the case of control systems, the function g defined by (25) follows to be only measurable with respect to t variable, so the notion of classical solution is not appropriate. We will use instead the notion of Carathéodory solution. Definition: Let Ω be an open set in R R n and let g : Ω R n be a function which define an equation like (24). A function ϕ : I R R n, with I an interval, is called a Carathéodory solution of (24) if ϕ AC (I, R n ), G (ϕ) := {(t, ϕ (t)) : t I } Ω and ϕ (t) = g (t, ϕ (t)) a.e t I. (26) Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 56 / 120

Carathéodory solutions: some remarks (i) An absolutely continuous function ϕ : [t 0, t 1 ] R n, is a Carathéodory solution of (24) if G (ϕ) := {(t, ϕ (t)) : t [t 0, t 1 ]} Ω and t x (t) = x (t 0 ) + g (s, x (s)) ds for every t t 0 [t 0, t 1 ]. (ii) Any classical solution is a Carathéodory solution (iii) The function ϕ : [ 1, 1] R defined by ϕ (t) = { t if t 0 t if t < 0 is a Carathéodory solution but not a classical solutions of x = 1, x (0) = 0. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 57 / 120

Existence of Carathéodory solutions We assume that the function g : Ω R R n R n satisfies the following conditions: (A) For every x the function t g (t, x) defined on the section Ω x = {t : (t, x) Ω} is measurable. For every t, the function x g (t, x) defined on the section Ω t = {x : (t, x) Ω} is continuous (B) For every compact K Ω there exist constants C K, L K such that g (t, x) C K, g (t, x) g (t, y) L K x y, for all (t, x), (t, y) K.. (27) For some (t 0, x 0 ) Ω we consider the Cauchy problem (g, t 0, x 0 ) : ẋ = g (t, x), x (t 0 ) = x 0. (28) Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 58 / 120

Existence of Carathéodory solutions Theorem (existence) Given a map g : Ω R n, consider the Cauchy problem (g, t 0, x 0 ) in (28) for some (t 0, x 0 ) Ω. (i) If g satisfies (A), (B), then there exists ε > 0 such that (28) has a local solution x (.) defined for t [t 0, t 0 + ε] (ii) Assume, in addition that the function g is defined in the entire space R R n and there exist constants C, L such that g (t, x) C, g (t, x) g (t, y) L x y (29) for all (t, x), (t, y) R R n. Then for every T > t 0, the Cauchy problem (28) has a global solution x (.) defined for all t [t 0, T ]. Moreover, the solution depends continuously on the initial data x 0. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 59 / 120

Proof of the Existence of Carathéodory solutions Assume that g satisfies (A) in R R n and (29). Let T > t 0. We shall construct a forward solution x = x x0 : [t 0, T ] R n. In view of applying Theorem 1 (the Banach contraction mapping principle) we define Λ = R n (the initial value x 0 R n plays here the role of a parameter). Let X be the space C ([t 0, T ], R n ) endowed with the weighted norm x + = sup e. 2Lt x (t), t [t 0,T ] which is equivalent to the usual C ([t 0, T ], R n ) norm defined by x = sup t [t0,t ] x (t),and define Φ : Λ X X by setting t Φ (x 0, w) (t) = x 0 + g (s, w (s)) ds, t [t 0, T ]. t 0 (30) Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 60 / 120

Proof of the Existence of Carathéodory solutions To prove that Φ is well defined, for each w X we need to show that the composite function s g (s, w (s)) is integrable. Let δ = T t 0. Given a function w X we consider the sequence of piecewise constant functions (w n ) n N with w n (t) = w ( t 0 + k δ ) [, t t 0 + k δ n n, t 0 + (k + 1) δ ], 0 k n 1 n By assumption (A), the maps t g (t, w n (t)) are all measurable. Moreover, (29) implies that lim g (t, w (t)) g (t, w n (t)) lim L w (t) w n (t) = 0, n n hence t g (t, w (t)) is measurable, being a pointwise limit of a sequence of measurable functions. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 61 / 120

Proof of the Existence of Carathéodory solutions Since g (s, w (s)) C for all s, it follows that the integral in (30) is well defined and depends continuously on t. Hence Φ is well defined and takes values inside X. The continuity of the map x 0 Φ (x 0, w) is obvious. To prove that w Φ (x 0, w) is a contraction in ( X,. + ), let w, w X and let By the definition of. + and (31), we have θ = w w +. (31) w (s) w (s) θe.2ls for all s [t 0, T ]. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 62 / 120

Proof of the Existence of Carathéodory solutions Moreover, by (29), for all t [t 0, T ], e. 2Lt Φ (x0, w (t)) Φ ( x 0, w (t) ) t = e 2Lt [ ( g (s, w (s)) g s, w (s) )] ds t 0 t e 2Lt L w (s) w (s) ds t 0 t e 2Lt Lθe 2Ls ds < δ t 0 2. Therefore, Φ (x 0, w) Φ ( x 0, w ) + 1 w w 2 +. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 63 / 120

Proof of the Existence of Carathéodory solutions We can now apply the Banach contraction mapping principle, obtaining the existence of a unique continuous mapping x 0 x x0 such that x x0 = Φ (x 0, x x0 ) that is, t x x0 (t) = x 0 + g (s, x x0 (s)) ds, t [t 0, T ]. t 0 This means that x x0 proof of (ii). is the required solutions, and this achieves the Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 64 / 120

Proof of the Existence of Carathéodory solutions To prove (i) concerning local existence of solution without the additional assumption (29), choose ε > 0 small enough so that the set K = {(t, x) : t t 0 ε, x x 0 ε} is entirely contained in Ω. Then consider a smooth cut-off function φ : R R n [0, 1] such that φ 1 on K, while φ 0 outside some larger compact set K, with K K Ω. Observe that the function { g + φ (t, x) g (t, x) if (t, x) Ω (t, x) = 0 if (t, x) / Ω satisfies (A) and (B), together with the extra assumption (29), because it vanishes outside the compact set K. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 65 / 120

Proof of the Existence of Carathéodory solutions By the previous steps, there exists a solution x (.) to the Cauchy problem x = g + (t, x), x (t 0 ) = x 0 defined on arbitrary large interval [t 0, T ]. We now recall that K is a neighborhood of (t 0, x 0 ). Therefore for some ε > 0 suffi ciently small, the point (t, x (t)) remains in K as t [t 0, t 0 + ε]. Since g and g + coincide on K, the function x (.) thus provides a local solution to the original problem (g, t 0, x 0 ) on the interval [t 0, t 0 + ε]. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 66 / 120

Uniqueness of Carathéodory solutions The next lemma represents a main ingredient in several uniqueness proofs and it is a simplified version of Gronwall s lemma: Lemma (Gronwall) Let z (.) be an absolutely continuous nonnegative function such that z (t 0 ) γ, ż (t) α (t) z (t) + β (t) for a.e. t [0, T ], (32) for some integrable functions α, β and some constant γ 0. Then z satisfies the inequality t t z (t) γe α(s)ds 0 + t t 0 t β (s) e s α(u)du ds for all t [0, T ]. (33) Theorem (uniqueness) Let g : Ω R n satisfies (A) and (B), as in the existence Theorem. Let x 1 (.), x 2 (.) be solutions of (28), defined on the intervals [t 0, t 1 ], [t 0, t 2 ] respectively. If T = min {t 1, t 2 }, then x 1 (t) = x 2 (t) for all t [t 0, T ]. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 67 / 120

Proof of the uniqueness of Carathéodory solutions Since g satisfies assumption (B), for the compact set K = {(t, x 1 (t)) : t [t 0, T ]} {(t, x 2 (t)) : t [t 0, T ]} there exists L K 0 such that g (t, x) g (t, y) L K x y for all (t, x), (t, y) K. For all t [t 0, T ] = [t 0, t 1 ] [t 0, t 2 ] let z (t) = x 1 (t) x 2 (t). One has that z (.) is absolutely continuous and ż (t) ẋ 1 (t) ẋ 2 (t) L K z (t) for almost all t [t 0, T ]. Applying Gronwall s lemma with α = L K, β = γ = 0 we obtain that z (t) 0 for all t [t 0, T ]. Hence z (t) = x 1 (t) x 2 (t) = 0 for all t [t 0, T ]. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 68 / 120

Characterization of maximal solutions Let g : Ω R R n R n. In general, a solution to the Cauchy problem ẋ = g (t, x), x (t 0 ) = x 0 (34) is defined only locally, for t in a neighborhood of the initial time t 0. If a solution cannot be extended beyong a certain time T, two cases may arise: (i) as t T, the point (t, x (t)) approaches the boundary Ω of the domain Ω; (ii) as t T, the solution blows up, that is x (t). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 69 / 120

Characterization of maximal solutions Theorem Let Ω R R n be an open set, g : Ω R n satisfies the basic assumptions (A) and (B), (t 0, x 0 ) Ω and let T = sup {τ : (34) has a solution defined on [t 0, τ]}. Then either T = + or lim t T [ x (t) + ] 1 = +, (35) d ((t, x (t)), Ω) where d ((t, x), Ω) = inf { t s + x y : (s, y) Ω}. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 70 / 120

Characterization of maximal solutions Proof: Assume that T <. If (35) does not hold, then there exist M > 0, ε > 0 and a sequence t n T as n, such that x (t n ) M, d ((t n, x (t n )), Ω) ε. By possibly taking a subsequence, we can assume that x (t n ) converges to a point x as n, with (T, x ) Ω. Let ρ > 0 so small that the cylinder is entirely contained in Ω. K = {(t, x) : t T ρ, x x 0 ρ} Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 71 / 120

Characterization of maximal solutions Consider a smooth cut-off function φ : R R n [0, 1] such that φ 1 on K, while φ 0 outside some larger compact set K, with K K Ω, and define g + (t, x) = { φ (t, x) g (t, x) if (t, x) Ω 0 if (t, x) / Ω (36) Then g + g on K, and in addition g + satisfies the global bounds C, L such that g + (t, x) C, g + (t, x) g + (t, y) L x y for all (t, x), (t, y) R R n, for some constants L, C 1. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 72 / 120

Characterization of maximal solutions Fix δ > 0 small enough so that (2C + 1) δ ρ, and choose n so large that x (t n ) x δ, T t n δ. By (ii) of the existence Theorem, the Cauchy problem y = g + (t, y), x (t n ) = x (t n ) has a solution y (.) defined on [t n, T + δ]. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 73 / 120

Characterization of maximal solutions We can now define an extension x of x by setting { x (t) if t0 t t x (t) = n y (t) if t n t T + δ. Since y (t) C for t [t n, T + δ], we have y (t) x y (t) x (t n ) + x (t n ) x C (t t n ) + δ 2C δ + δ ρ. Therefore, for all t [t n, T + δ], the point (t, y (t)) remains inside the compact K where g and g + coincide. The function x (.) thus provide a solution of the original problem (34), defined on a strictly larger interval [t 0, T + δ]. This contradicts the maximality of T, thus proving the theorem. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 74 / 120

Nonlinear control systems We consider the control system x = f (t, x, u), u (.) U (37) where the set of admissible controls is defined as U := {u : R R m : u measurable, u (t) U for all t} (38) We shall assume the following basic hypothesis: (H) : The set U R m of control values is compact, Ω R R n is an open set, the function f : Ω U R n is continuous in all variables and continuously differentiable with respect to x. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 75 / 120

Trajectory of nonlinear control systems Definition: An absolutely continuous function x : [a, b] R n is said to be a solution (or trajectory) of (37) if its graph {(t, x (t)) : t [0, T ]} is entirely contained in Ω, and if there exists a measurable control u : [a, b] U such that x (t) = f (t, x (t), u (t)) for almost every t [a, b]. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 76 / 120

An equivalent differential inclusion In connection with (37), define the multifunction F : Ω P (R n ) by and consider the differential inclusion F (t, x) = {f (t, x, ω) : ω U} (39) x F (t, x). (40) Definition: We call solution or trajectory of the differential inclusion (40) any absolutely continuous function x : [a, b] R n whose graph {(t, x (t)) : t [a, b]} is entirely contained in Ω, and x (t) F (t, x (t)) for almost every t [a, b]. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 77 / 120

An equivalent differential inclusion Observe that, for each (t, x), the set of admissible velocities x in (37) is given in parametrized form, as the image of a fixed set U R n. On the other hand, when we study the differential inclusion (40), we do not assume any parametrization. The next result due to Filippov, shows that the two approaches are essentially equivalent. Filippov s lemma An absolutely continuous function x : [a, b] R n is a trajectory of the control system (37) if and only if it is a trajectory of the differential inclusion (40) with F (t, x) defined by (39). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 78 / 120

Proof of Filippov s lemma If x : [a, b] R n is a trajectory of the control system (37) then its graph is entirely contained in Ω, and if there exists a measurable control u : [a, b] U such that Then x (t) = f (t, x (t), u (t)) for almost every t [a, b]. x (t) = f (t, x (t), u (t)) F (t, x (t)) for almost every t [a, b], hence it is a trajectory of the differential inclusion (40). Assume now that x : [a, b] R n is a trajectory of the differential inclusion (40). Then x (t) F (t, x (t)) for almost every t [a, b]. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 79 / 120

Proof of Filippov s lemma Fix any ω U and define the multifunction W : [a, b] 2 Rn by { {ω U : f (t, x (t), ω) = x W (t) = (t)} if x (t) F (t, x (t)) {ω} otherwise. Remark that W (t) is a compact subset and W (t) = {ω} for t in a set of null Lebesgue measure. Let u (t) be the first element of W (t). Then by Proposition 2 of Section 2 one has that t u (t) is measurable, u (t) W (t) hence x (t) = f (t, x (t), u (t)), and x (.) is a trajectory of the control system (37). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 80 / 120

Fundamental properties of trajectories Let f, U satisfy the basic assumption (H) and let (0, x) Ω. For any measurable control u : [0, T ] U, the Cauchy problem x = f (t, x, u (t)), x (0) = x (41) is equivalent to x = g (t, x), x (0) = x. (42) where g (t, x) = f (t, x, u (t)) is suffi ciently regular to have a local solution x (., u) : [0, δ] R n of (42). Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 81 / 120

Fundamental properties of trajectories To study how the solution x (., u) depends on u, we first consider the globally bounded case, assuming (H ) : The set U R m of control values is compact, the function f : R R n U R n is continuous in all variables and continuously differentiable with respect to x and satisfies f (t, x, ω) C, D x f (t, x, ω) L, (43) for some constants C, L 0 and all (t, x, ω) R R n U. Then we have the following global existence and continuous dependence result: Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 82 / 120

Fundamental properties of trajectories Theorem (imput-output map): Let the assumption (H ) be satisfied. Then, for every T > 0 and u U the Cauchy problem (41) has a unique solution x (., u) : [0, T ] R n. The imput-output map u x (., u) is continuous from L 1 ([0, T ], R m ) to C ([0, T ], R n ) (the space of continuous functions from [0, T ] into R n. The continuous dependence of trajectories on the control is a useful basic result. Consider now a sequence of admissible controls (u n ) n N such that the sequence of the corresponding solutions (x (., u n )) n N converge to x (.) uniformly on [0, T ]. A natural question is whether the function x is a solution of the original problem. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 83 / 120

Fundamental properties of trajectories Example: Consider the system and for t R define x (t) = u (t), x (0) = 0, u (t) { 1, 1} a.e. (44) u n (t) = { 1 if sin (nt) 0 1 if sin (nt) < 0. Then x (., u n ) converges to x 0 uniformly for all t R but x 0 is not a trajectory of (44). We can conclude that the set of trajectories of (44) is not closed. Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018 84 / 120