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Lasalle invariance principle for ordinary dierential equations and applications

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Table Of Contents

Acknowledgment i Certication ii Approval iii Introduction v Dedication vi 1 Preliminaries 2
1.1Denitions and basic Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2Exponential of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Basic Theory of Ordinary Dierential Equations 7
2.1Denitions and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2Continuous dependence with respect to the initial conditions . . . . . . . . . . . . 11
2.3Local existence and blowing up phenomena for ODEs . . . . . . . . . . . . . . . . 12
2.4Variation of constants formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Stability via linearization principle 21
3.1Denitions and basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Lyapunov functions and LaSalle’s invariance principle 26
4.1Denitions and basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2Instability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3How to search for a Lyapunov function (variable gradient method) . . . . . . . . . 30
4.4LaSalle’s invariance principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5Barbashin and Krasorskii Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6Linear systems and linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5 More applications 40
5.1Control design based on lyapunov’s direct method . . . . . . . . . . . . . . . . . . 41 Conclusion 48 Bibliography 48

Thesis Abstract

Abstract
The LaSalle invariance principle is a fundamental tool in the analysis of stability for systems described by ordinary differential equations. This principle provides a powerful method for proving the asymptotic stability of equilibrium points or limit cycles. The essence of the LaSalle invariance principle lies in constructing a Lyapunov function that decreases along solutions of the differential equation, eventually leading to the convergence of trajectories to a region called the largest invariant set. This region typically corresponds to the set of points where the derivative of the Lyapunov function is zero. The LaSalle invariance principle has a wide range of applications in various fields, including control theory, dynamical systems, and mathematical biology. In this research project, we delve into the theoretical foundations of the LaSalle invariance principle and explore its applications in the context of ordinary differential equations. We study the conditions under which the LaSalle invariance principle can be applied to guarantee stability properties of solutions to differential equations. By analyzing the behavior of trajectories near equilibrium points or limit cycles, we aim to establish the asymptotic stability of these critical points using Lyapunov functions constructed based on the LaSalle invariance principle. Furthermore, we investigate the robustness of the LaSalle invariance principle with respect to perturbations in the system dynamics. Understanding how small changes in the system parameters affect the stability analysis is crucial for practical applications of the principle. We explore techniques for incorporating uncertainties into the stability analysis framework and develop methods for quantifying the impact of perturbations on the stability properties of the system. Moreover, we consider practical examples to illustrate the effectiveness of the LaSalle invariance principle in analyzing stability properties of real-world systems. By applying the principle to specific differential equations modeling physical, biological, or engineering systems, we demonstrate how it can be used to predict the long-term behavior of trajectories and assess the stability of the system under different operating conditions. Overall, this research project contributes to the understanding and application of the LaSalle invariance principle for ordinary differential equations, shedding light on its significance in stability analysis and its relevance to a wide range of scientific and engineering disciplines.

Thesis Overview

PRELIMINARIES 1.1 Denitions and basic Theorems In this chapter, we focussed on the basic concepts of the ordinary dierential equations. Also, we emphasized on relevant theroems in ordinary dierential equations. Denition 1.1.1 An equation containing only ordinary derivatives of one or more dependent vari- ables with respect to a single independent variable is called an ordinary dierential equation ODE. The order of an ODE is the order of the highest derivative in the equation. In symbol, we can express an n-th order ODE by the form x(n) = f(t; x; :::; x(nô€€€1)) (1.1.1) Denition 1.1.2 (Autonomous ODE ) When f is time-independent, then (1.1.1) is said to be an autonomous ODE. For example, x0(t) = sin(x(t)) Denition 1.1.3 (Non-autonomous ODE ) When f is time-dependent, then (1.1.1) is said to be a non autonomous ODE. For example, x0(t) = (1 + t2)y2(t) Denition 1.1.4 f : Rn ! Rn is said to be locally Lipschitz, if for all r > 0 there exists k(r) > 0 such that kf(x) ô€€€ f(y)k k(r)kx ô€€€ yk; for all x; y 2 B(0; r): f : Rn ! Rn is said to be Lipschitz, if there exists k > 0 such that kf(x) ô€€€ f(y)k kkx ô€€€ yk; for all x; y 2 Rn: Denition 1.1.5 (Initial value problem (IVP) Let I be an interval containing x0, the follow- ing problem ( x(n)(t) = f(t; x(t); :::; x(nô€€€1)(t)) x(t0) = x0; x0(t0) = x1; :::; x(nô€€€1)(t0) = xnô€€€1 (1.1.2) is called an initial value problem (IVP). x(t0) = x0; x0(t0) = x1; :::; x(nô€€€1)(t0) = xnô€€€1 are called initial condition. 2 Lemma 1.1.6 [9](Gronwall’s Lemma) Let u; v : [a; b] ! R+ be continuous such that there exists > 0 such that u(x) +x a u(s)v(s)ds; for all x 2 [a; b]: Then, u(x) ex a v(s)ds ; for all x 2 [a; b]: Proof . u(x) +x a u(s)v(s)ds implies that u(x) +x a u(s)v(s)ds v(x): So, u(x)v(x) +x a u(s)v(s)ds v(x); which implies that x a u(x)v(x) +x a u(s)v(s)ds dsx a v(x)ds: So, taking exponential of both side we get u(x) +x a u(s)v(s)dsx a u(s)v(s)ds: Thus, u(x) ex a v(s)ds ; x 2 [a; b]: Corollary 1.1.7 Let u; v : [a; b] ! R+ be continuous such that u(x)x a u(s)v(s)ds; for all x 2 [a; b]: Then, u = 0 on [a; b]. Proof . Now, u(x)x a u(s)v(s)ds implies that u(x)x a u(s)v(s)ds 1 n +x a u(s)v(s)ds; for all n 1: So, by Gronwall’s lemma, u(x) 1 n ex a u(s)v(s)ds; so as n ! 1; u(x) ! 0: Thus, u(x) = 0, since u(x) 0. Hence, u = 0 on [a; b]. 3 1.2 Exponential of matrices Denition 1.2.1 Let A 2 Mnn(R), then eA is an n n matrix given by the power series eA = 1X k=0 Ak k! The series above converges absolutely for all A 2 Mnn(R) Proof . The n-th partial sum is Sn = Xn k=0 Ak k! So, let n > m Then, Sn ô€€€ Sm = Xn k=m+1 Ak k! : So, kSn ô€€€ Smk Xn k=m+1 kAkk k! : So as m ! 1; kSn ô€€€ Smk ! 0 So, (Sn)n is Cauchy. Thus, converges. Theorem 1.2.2 [3](Cayley Hamilton Theorem) Let A 2 Mnn(R) and () = det(I ô€€€ A) its characteristic polynomial then (A) = 0: Proof . Let A 2 Mnn(R); () = det(I ô€€€ A) = c0 + c1 + c22 + ::: + cnn: adj(A ô€€€ I) = B0 + B1 + B22 + ::: + Bnô€€€2nô€€€2 + Bnô€€€1nô€€€1; where Bi 2 Mnn(R) for i = 0; 1; 2; :::; n; but, from linear algebra we have that Aô€€€1 = adj(A) det(A) ; where adj(A) denotes the adjugate or classical adjoint of A. So, det(I ô€€€ tA)I = (I ô€€€ tA)adj(I ô€€€ tA): (A ô€€€ I)(B0 + B1 + B22 + ::: + Bnô€€€2nô€€€2 + Bnô€€€1nô€€€1) = (c0 + c1 + c22 + ::: + cnn)I: Observe that the entries in adj(I ô€€€tA) are polynomials in of degree at most nô€€€1. So, Bi is the zero matrix for i = n. Equating the coecients of n on both sides gives c0I + c1A + c2A2 + ::: + cnAn = 0: Thus, (A) = 0: 4 Example 1.2.3 (Application of Cayley Hamilton Theorem) Find etA for A =0 1 ô€€€1 0 ! Solution: The characteristic equation is s2 + 1 = 0, and the eigenvalues are 1 = i, and 2 = ô€€€i. So, by Theorem 1.2.2 we have that, etA = 0I + 1A; where we are to nd the value of 0, and 1. So, eti = cos t + i sin t = 0 + 1i eô€€€ti = cos t ô€€€ i sin t = 0 ô€€€ 1i which implies that 0 = cos t, and 1 = sin t. So, etA = cos(t)I + sin(t)A =cos t sin t ô€€€sin t cos t ! Theorem 1.2.4 [11] Let A;B 2 Mnn(R). Then, (1) If 0 denotes the zero matrix, then e0 = I, the identity matrix. (2) If A is invertible, then eABAô€€€1 = AeBAô€€€1. Proof . Recall that, for all integers s 0, we have (ABAô€€€1)s = ABsAô€€€1. Now, eABAô€€€1 = I + ABAô€€€1 + (ABAô€€€1)2 2! + ::: = I + ABAô€€€1 + AB2Aô€€€1 2! + ::: = A(I + B + B2 2! + :::)Aô€€€1 = AeBAô€€€1: (3) If A is symmetric such that A = AT , then e(AT ) = (eA)T : Proof . eA = 1X k=0 Ak k! : Then eAT = 1X k=0 (AT )k k! = 1X k=0 (Ak)T k! = ( 1X k=0 Ak k! )T = (eA)T : (4) If AB = BA, then eA+B = eAeB: Proof . eAeB = (I + A + A2 2! + A3 3! + :::)(I + B + B2 2! + B3 3! + :::) = ( 1X k=0 Ak k! )( 1X j=0 Bj j! ) = 1X k=0 1X j=0 (A + B)k+j j!k! 5 Put m = j + k, then j = m ô€€€ k then from the binomial theorem that eAeB = 1X m=0 1X k=0 AmBmô€€€k (m ô€€€ k)!k! = 1X m=0 Am m! 1X k=0 m! (m ô€€€ k)! Bmô€€€k k! = 1X m=0 (A + B)m m! = eA+B: Theorem 1.2.5 [9] detA dt = AetA = etAA; for t 2 R: Proof . x(t; x0) = etAx0. Then, dx(t; x0) dt = etAx0A = 1X k=0 tkAk k! x0A = ( lim n!1 Xn k=0 tkAk k! )x0A = lim n!1 Xn k=0 tkAk+1 k! x0 = lim n!1 Xn k=0 AtkAk k! x0 = A 1X k=0 tkAk k! x0 = AetAx0 So, detA dt = AetA = etAA: Proposition 1.2.6 The solution x(:; x0) of the following linear space ( x0(t) = Ax(t); t 2 R x(0) = x0 2 Rn where A 2 Mnn(R), is given by x(t; x0) = etAx0: 6

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Lasalle invariance principle for ordinary dierential equations and applications

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