Matrices in Action

Linear dynamical systems are defined in this section, highlighting their properties and relevant matrix functions Additionally, Section A.1 of the Appendix summarizes the essential matrix algebra notation, terminology, and fundamental concepts necessary for understanding these systems.

A For instance, a statement in which the symbolFappears is supposed to be true whetherFis the real fieldRor the complex fieldC.

A dynamical system on \( F^n \) is defined as a triple \( (F^n, T, \Theta) \), where \( F^n \) represents the state space and elements \( x \in F^n \) are referred to as states The time-set \( T \) can be either \( R^+ \) or \( Z^+ \) The transition mapping, known as the evolution function \( \Theta_t \), determines the state at time \( t \) based on the initial state \( x(0) := \xi \in F^n \), satisfying the conditions \( x(t) = \Theta_t(\xi) \) and \( \Theta_0(\xi) = \xi \) Additionally, it follows the property \( \Theta_s(\Theta_t(\xi)) = \Theta_{s+t}(\xi) \) for all \( s, t \in T \) and \( \xi \in F^n \) A dynamical system is classified as linear if it holds that for all \( x, z \in F^n \), \( t \in T \), and scalars \( \alpha, \beta \in F \), the equation \( \Theta_t(\alpha x + \beta z) = \alpha \Theta_t(x) + \beta \Theta_t(z) \) is satisfied.

In this definition, if T := Z+ = {0,1,2, } we call the triple adiscrete- time dynamical system;its mappingsΘ t can be obtained from the mapping

1.1 Matrices in Action 3 Θ1 :F n →F n by using (1.1).Discrete-time linear dynamical systemsonF n have a transition mappingΘ t that satisfies (1.2); then there exists a square matrix

Asuch thatΘ1(x)=Ax; x(t+1)=Ax(t), or succinctlyx + =Ax; Θ t (ξ)=A t ξ (1.3)

A continuous-time dynamical system is defined by a differentiable function Θt in the variables x and t, represented as IfT = R+ The transition mapping can be modeled as a semi-flow Θ : R+ × F n → F n, where the unique solution x(t) = Θt(ξ) is continuous and defined for all R+ This solution arises from a first-order differential equation ˙x = f(x) with the initial condition x(0) = ξ, which characterizes the system's dynamics.

If instead of the semi-flow assumption we postulate that the transition mapping satisfies (1.2) (linearity) then the triple is called acontinuous-time linear dynamical systemand lim t↓0

1 t(Θ t (x)−x)=Ax for some matrixA∈F n×n From that and (1.1), it follows that the mapping x(t)=Θ t (x) is a semi-flow generated by thedynamics ˙ x=Ax (1.4)

Given an initial statex(0) :=ξsetx(t) :=X(t)ξ; then (1.4) reduces to a single initial-value problem for matrices: find ann×nmatrix functionX:R+ →

Proposition 1.1 The initial-value problem(1.5)has a unique solution X(t)onR+ for any A∈F n×n

Proof The formal power series int

2 +ã ã ã+A k t k k!+ã ã ã (1.6) and its term-by-term derivative ˙X(t) formally satisfy (1.5) Using the inequal- ities (see (A.3) in Appendix A) satisfied by the operator normã, we see that on any finite time interval [0,T]

Section B.3.2 discusses nonlinear dynamical systems characterized by finite escape time, where equation (1.1) is valid only for sufficiently small values of s + t Consequently, equation (1.6) demonstrates uniform convergence on the interval [0, T] The series representing the derivative ˙X(t) also converges uniformly, confirming that X(t) is a solution to equation (1.5) Furthermore, the series for higher derivatives converge uniformly on [0, T], indicating that X(t) possesses derivatives of all orders.

The uniqueness ofX(t) can be seen by comparison with any other solution Y(t) Let

As a corollary, givenξ∈F n the unique solution of the initial-value prob- lem for (1.4) givenξis the transition mappingΘ t x=X(t)x.

Definition 1.2.Define thematrix exponential functionby e tA =X(t), whereX(t) is the solution of (1.5) △

The transition mapping for the equation (1.4) is represented by Θ t (x) exp(tA)x, while equation (1.1) indicates that the matrix exponential function satisfies the property e tA e sA = e (t + s)A (1.7), which can also be derived from the series in (1.6); this derivation is suggested as an exercise in series manipulation Various methods for calculating the matrix exponential function exp(A) are detailed in the work of Moler and Van Loan [211].

Exercise 1.1.For anyA∈F n×n and integerk≥n,A k is a polynomial inAof degreen−1 or less (see Appendix A) △

Matrix functions, such as exp(A), can be defined through various methods, including solving matrix differential equations and utilizing power series The eigenvalues of a matrix A, denoted as spec(A) = {α₁, , αₙ}, play a crucial role in this context If a function ψ(z) has a Taylor series that converges within a disk U:={z | |z| 0 \) for all nonzero vectors \( x \) Conversely, \( Q \) is negative definite, represented as \( Q \ll 0 \), if \( n_Q = n \).

Certain of thekth-orderminorsof a matrixQ(Section A.1.2) are itsleading principal minors, the determinants

Proposition 1.3 (J J Sylvester) Suppose Q ∗ = Q, then Q ≫ 0 if and only if

Proofs are given in Gantmacher [101, Ch X, Th 3] and Horn and Johnson [131].

If spec(A)={α1, , α n }the linear operator

Ly A : Symm(n)→Symm(n), Ly A (Q) :=A ⊤ Q+QA, called theLyapunov operator, has then 2 eigenvaluesα i +α j , 1 ≤ i,j≤ n, so

Ly A (Q) is invertible if and only if all the sumsα i +α j are non-zero.

Proposition 1.4 (A M Lyapunov) The real matrix A is a Hurwitz matrix if and only if there exist matrices P,Q∈Symm(n)such that Q≫0, P≫0and

For proofs of this proposition, see Gantmacher [101, Ch XV], Hahn [113,

Ch 4], or Horn and Johnson [132, 2.2].

The test in Proposition 1.4 fails if any eigenvalue is purely imaginary; but even then, ifPbelongs to the range of Ly A then (1.17) has solutionsQwith z Q >0.

TheLyapunov equation(1.17) is a special case ofAX+XB=C, theSylvester equation; the operatorX → AX+XB, where A,B,X ∈ F n×n , is called the Sylvester operator 5 and is discussed in Section A.3.3.

If \( A \in \mathbb{R}^{n \times n} \), its complex eigenvalues appear in conjugate pairs For any similar matrix \( C = S^{-1}AS \in \mathbb{C}^n \), including those in triangular or Jordan canonical form, both \( C \) and \( A \) are classified as Hurwitz matrices This classification holds true if there exists a Hermitian matrix \( P \gg 0 \) such that the Lyapunov equation \( C^*Q + QC = -P \) has a Hermitian solution \( Q \gg 0 \).

Linear Control Systems

Linear control systems serve as a compelling example to establish foundational definitions that can be easily expanded upon Unlike dynamical systems, control systems offer the freedom of choice, allowing for multiple solutions from a given initial state For instance, vehicles can be viewed as control systems, guided by humans or computers to achieve specific orientations and positions in two-dimensional or three-dimensional space.

The Sylvester equation is explored in several key texts, including works by Bellman, Gantmacher, and Horn and Johnson For numerical solutions, software such as MATLAB and Mathematica can be utilized, with the most recognized algorithm developed by Bartels and Stewart.

6 A general concept of control system is treated by Sontag [249, Ch 2] There are many books on linear control systems suitable for engineering students, including Kailath [152] and Corless and Frazho [65].

A continuous-time linear control system is defined as a quadruple (F n, R+, U, Θ), where F n represents the state space, R+ denotes the time-set, and U is a class of input functions (controls) mapping from R+ to R m The transition mapping Θ is parameterized by input functions u from U, generated by the controlled dynamics described by the equation ˙x(t) = Ax(t) + Bu(t) Additionally, the control system may include an output y(t) = Cx(t) as part of its description It is assumed that the system is time-invariant, meaning the coefficient matrices A, B, and C remain constant.

The largest class of functions we require is LI, which consists of R m-valued locally integrable functions These functions are characterized by the existence of the Lebesgue integral s t u(τ)dτ for all 0≤ s ≤t < ∞ To derive an explicit transition mapping, we can utilize the fact that for any time interval [0,T], a control u from the LI class and an initial state x(0)=ξ uniquely determine the solution of the differential equation, represented as x(t)=e tA ξ+ t.

The specification of the control system commonly includes a control constraint u(ã) ∈ Ω where Ω is a closed set Examples are Ω := R m or Ω := {u||u i (t)| ≤1,1≤i≤m} 8 The input function spaceU ⊂ LI must be invariant under time-shifts.

Definition 1.4 (Shifts).The function vobtained by shifting a functionuto the right along the time axis by a finite timeσ≥0 is writtenv =S σ uwhere

S ( ã ) is thetime-shiftoperator defined as follows If the domain ofuis [0,T], then

The space of piecewise constant functions, denoted as PK, is a shift-invariant subspace of LI, characterized by the property that for any interval [0,T], the function u takes constant values in R^m across all open intervals of a partition of [0,T], with no requirement for definition at the endpoints Additionally, another shift-invariant subspace of LI, referred to as PC[0,T], consists of R^m-valued piecewise continuous functions on the interval [0,T], where it is assumed that the limits at the endpoints of the pieces are finite.

With an outputy=Cxandξ=0 in (1.19), the relation between input and output becomes a linear mapping

7 State spaces linear over C are needed for computations with triangularizations through- out, and in Section 6.5 for quantum mechanical systems.

8 Less usually, Ω may be a finite set, for instance Ω = {−1, 1}, as in Section 6.3. y=L u where L u(t) :=C t

Since the coefficientsA,B,Care constant, it is easily checked that S σ L(u) LS σ (u) Such an operatorListime-invariant;Lis alsocausal:L u(T) depends only on the input’s pasthistoryU T :={u(s),s0 thet-attainable set isA t (ξ)={x|x=ξe t }, a circle, so (1.47) does not have the strong accessibility property anywhere The attainable set is

The set A(ξ) = {x | x > ξ} is not controllable in R²*, but it does possess an open interior, indicating that while it is neither fully open nor closed, it meets the accessibility property In the R² × R⁺ space-time framework, the set {(Aₜ(ξ), t) | t > 0} forms a truncated circular cone.

In the modified system where the roles of I and J are reversed, we express ˙x as (J + uI)x, leading to polar coordinates represented by ρ˙ = uρ and θ˙ = 1 At any time T > 0, the radius ρ can take any positive value, while the angle θ(T) is given by 2πT + θ0 This system exhibits the accessibility property, though it is not strongly accessible, and is controllable, allowing for the desired final angle to be achieved only once per second The trajectories of this system in space-time are characterized by the set {(t, u, θ0 + 2πt, t) | t ≥ 0}, which reside on the positive half of a helicoid surface.

A matrix semigroupS⊂R n×n is said to to betransitive onR n ∗ if for every pair of states{x,y}there exists at least one matrixA∈Ssuch thatAx=y.

22 The discrete-time system x(t + 1) = u(t)x(t) is controllable on R 1 ∗

23 In Example 1.8, the helicoid surface {(r, 2πt, t) | t ∈ R, r > 0} covers R 2 ∗ exactly the way the Riemann surface for log(z) covers C ∗ , with fundamental group Z.

In a controllable bilinear system, the semigroup of transition matrices, denoted as S Ω, exhibits transitivity on R n ∗ This concept parallels the transition matrices of discrete-time systems, which will be explored in Chapter 4 For symmetric systems, controllability is characterized by the transitivity of the matrix group Φ on R n ∗ In systems with drift, establishing controllability typically involves demonstrating that S Ω functions as a transitive group.

Example 1.9.An example of a semigroup (without identity) of matrices tran- sitive onR n ∗ is the set of matrices of rankk,

The set \( S_n^k \) is defined as \( \{ A \in \mathbb{R}^{n \times n} | \text{rank} A = k \} \) These sets form semigroups; however, for \( k < n \), they lack an identity element To demonstrate the transitivity of \( S_n^1 \), consider \( x, y \in \mathbb{R}^{n*} \) and let \( A = \frac{yx^\top}{x^\top x} \), which implies \( Ax = y \) For \( k > 1 \), these semigroups satisfy \( S_n^k \supset S_n^{k-1} \) and are transitive on \( \mathbb{R}^{n*} \), with the largest being \( S_n^n = GL(n, \mathbb{R}) \).

Example 1.10.Transitivity has many contexts ForA,B∈GL(n,R), there exists

X∈GL(n,R) such thatXA=B The columns ofA=(a 1, ,a n ) are a linearly independentn-tuple of vectors Such an n-tuple Ais called ann -frameon

R n ; so GL(n,R) is transitive on n-frames For n-frames on differentiable n-manifolds, see Conlon [64, Sec 5.5] △

Stability: Nonlinear Dynamics

In succeeding chapters, stability questions fornonlineardynamical systems will arise, especially in the context of state-dependent controlsu=φ(x) The best-known tool for attacking such questions is Lyapunov’s direct method.

In this section, our dynamical system has state space R n with transition mapping generated by a differential equation ˙x = f(x), assuming that f satisfies conditions guaranteeing that for everyξ∈R n there exists a unique trajectory inR n x(t)= f t (ξ), 0≤t≤ ∞.

To identify the equilibrium states \( x_e \) that meet the condition \( f(x_e) = 0 \), we recognize that the trajectory \( f_t(x_e) = x_e \) represents a constant point For a more in-depth analysis of an isolated equilibrium state \( x_e \), we can simplify our approach by transforming the coordinates, positioning \( x_e \) at the origin (0) This will serve as our foundational assumption for further investigation.

The state 0 is calledstableif there exists a neighborhoodUof 0 on which f t (ξ) is continuousuniformly for t∈R+ If also f t (ξ)→0 ast→ ∞for allξ∈U, the equilibrium isasymptotically stable 25 onU To show stability (alternatively,

24 Lyapunov’s direct method can be used to investigate the stability of more general invariant sets of dynamical systems.

25 The linear dynamical system ˙ x = Ax is asymptotically stable at 0 if and only if A is aHurwitz matrix.

In the study of nonlinear dynamics and stability, one can construct a family of disjoint, compact, and connected hypersurfaces containing the origin, ensuring that all trajectories remain within these surfaces A straightforward method to achieve this is by utilizing the level sets {x | V(x) = δ} of a continuous function V: R^n → R, where V(0) = 0 and V(x) > 0 for x ≠ 0 This function V is a generalization of the significant example x⊤Qx, where Q is a positive definite matrix, and is referred to as positive definite (V ≫ 0) Definitions also exist for positive semidefinite (V(x) ≥ 0, x ≠ 0) and negative semidefinite (V(x) ≤ 0, x ≠ 0), while if (-V) is positive definite, it is denoted as V ≪ 0 (negative definite).

A gauge function is defined as a C^1 function that is positive definite in a neighborhood around the origin By assuming that each level set forms a compact hypersurface, it is possible to construct regions and boundaries for δ > 0.

A gauge function is said to be a Lyapunov function for the stability of f t (0)=0, if there exists aδ >0 such that ifξ∈Vδandt>0,

V( f t (ξ))≤V(ξ), that is, f t (∂Vδ)⊆Vδ. One way to prove asymptotic stability uses stronger inequalities: if fort>0 V( f t (ξ))0, onVδ both V(x)>0and f V(x)