9780387307695-c2 (1).pdf - 2 Linear Systems and Stability...

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2 Linear Systems and Stability of Nonlinear Systems In this chapter we will study the differential equation ˙ x = A ( t ) x + f ( x, t ) , x R n where A is a smooth n × n matrix-valued function and f is a smooth function such that f (0 , t ) = f x (0 , t ) 0. Note that if f has this form, then the associated homogeneous linear system ˙ x = A ( t ) x is the linearization of the differential equation along the zero solution t φ ( t ) 0. One of the main objectives of the chapter is the proof of the basic results related to the principle of linearized stability. For example, we will prove that if the matrix A is constant and all of its eigenvalues have negative real parts, then the zero solution (also called the trivial solution ) is asymptoti- cally stable. Much of the chapter, however, is devoted to the general theory of homogeneous linear systems; that is, systems of the form ˙ x = A ( t ) x . In particular, we will study the important special cases where A is a constant or periodic function. In case t A ( t ) is a constant function, we will show how to reduce the solution of the system ˙ x = Ax to a problem in linear algebra. Also, by defining the matrix exponential, we will discuss the flow of this autonomous system as a one-parameter group with generator A . Although the behavior of the general nonautonomous system ˙ x = A ( t ) x is not completely understood, the special case where t A ( t ) is a periodic matrix-valued function is reducible to the constant matrix case. We will develop a useful theory of periodic matrix systems, called Floquet theory, and use it to prove this basic result. The Floquet theory will appear again later when we discuss the stability of periodic nonhomogeneous systems. In
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146 2. Linear Systems and Stability of Nonlinear Systems particular, we will use Floquet theory in a stability analysis of the inverted pendulum (see Section 3.5). Because linear systems theory is so well developed, it is used extensively in many areas of applied science. For example, linear systems theory is an essential tool for electromagnetics, circuit theory, and the theory of vibra- tion. In addition, the results of this chapter are a fundamental component of control theory. 2.1 Homogeneous Linear Differential Equations This section is devoted to a general discussion of the homogeneous linear system ˙ x = A ( t ) x, x R n where t A ( t ) is a smooth function from some open interval J R to the space of n × n matrices. Here, the continuity properties of matrix-valued functions are determined by viewing the space of n × n matrices as R n 2 ; that is, every matrix is viewed as an element in the Cartesian space by simply listing the rows of the matrix consecutively to form a row vector of length n 2 . We will prove an important general inequality and then use it to show that solutions of linear systems cannot blow up in finite time. We will discuss the basic result that the set of solutions of a linear system is a vector space, and we will exploit this fact by showing how to construct the general solution of a linear homogeneous system with constant coefficients.
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