##
**Practical stability of nonlinear systems.**
*(English)*
Zbl 0753.34037

Singapore: World Scientific,. x, 207 p. (1990).

From a practical point of view, a technical system will be considered stable if the deviations of the motion from the desired state remain within certain bounds determined by the physical situation, in case the initial values and disturbances are bounded by suitable constants. This concept of “practical stability” (which is not implied by and does not imply Lyapunov stability) was already formulated by J. P. La Salle and S. Lefschetz [Stability by Lyapunov’s direct method with applications. New York (1961; Zbl 0098.061)] and studied by many authors. In particular, important contributions have been given by the third author over the last 20 years. This monograph presents a systematic account of the development, describes the current state of the theory and provides a unified general approach to a broad variety of nonlinear problems.

The book consists of four chapters. Chapter 1 starts with definitions: a system (1) \(x'=f(t,x)\) where \(f\in C[\mathbb{R}_ +\times\mathbb{R}^ n,\mathbb{R}^ n]\) is said to be “practically stable if, given \((\lambda,A)\) with \(0<\lambda<A\), we have \(| x_ 0|<\lambda\) implies \(| x(t)|<A\), \(t\geq t_ 0\) for some \(t_ 0\in\mathbb{R}_ +\)”, where, as usual, \(x(t)\) denotes the solution of (1) with initial value \(x(t_ 0)=x_ 0\). {The reviewer has to confess that this definition seems to him to be somewhat fuzzy. What do the words “given \((\lambda,A)\)” really mean? Of course the decision whether or not a particular system is practically stable strongly depends on the chosen values of \(\lambda\) and \(A\), and so the notion, say, “\((\lambda,A)\)-practically stable” would be more appropriate.} In addition to the definition above, 11 modifications are given, culminating in the notion “eventually uniformly strongly practically stable”. After that, criteria for practical stability are presented, using scalar comparison systems (2) \(u'=g(t,u)\) where \(u\in C(\mathbb{R}_ +\times\mathbb{R}_ +,\mathbb{R})\), and stating conditions under which the practical stability of (2) implies practical stability of (1). Analogous criteria are also given for delay differential systems, integro-differential equations, difference equations, and impulsive differential equations.

In chapter 2, Lyapunov functions are introduced as additional tools for investigating practical stability. As usual, the Lyapunov functions are assumed to satisfy an inclusion \(a(| x|)\leq V(x,t)\leq b(| x|)\) with non-decreasing functions \(a(r)\), \(b(r)\); in addition, these functions now have to satisfy the inequality \(a(\lambda)<b(A)\). More general approaches using perturbing Lyapunov functions or vector Lyapunov functions are discussed. For large-scale systems a decomposition- aggregation method is presented. Then, inspired by the geometrical interpretation of the Lyapunov function level sets, more general definitions of practical stability in terms of arbitrary sets (i.e., \(x_ 0\in S_ 0(t_ 0)\) implies \(x(t)\in S(t)\) for \(t\geq t_ 0)\) or in terms of two measures (i.e., \(h_ 0(t_ 0,x_ 0)<\lambda\) implies \(h(t,x)<A\) for \(t\geq t_ 0)\) are introduced, and also with respect to these new notions stability criteria are obtained.

Chapter 3 is devoted to perturbed systems (3) \(x'=f(t,x)+R(t,x)\). The notion of practical stability under constantly acting perturbations (i.e., \(x_ 0\in S_ 0(t_ 0)\) and \(R(t,x)\in P\) for \(x\in S(t)\), \(t\geq t_ 0\) imply \(x(t)\in S(t)\) for \(t\geq t_ 0)\) and a new comparison theorem are presented, and again a large number of stability criteria, for systems (3) as well as for various other types of nonlinear systems, are obtained.

In chapter 4, control systems (4) \(x'=f(t,x,n)\) are considered. Using a Lyapunov function, conditions are formulated which assure that the practical stability properties of (4) are determined by the corresponding properties of a scalar comparison equation. {Note that this comparison equation should read \(w'=g(t,w,v)\). Equation (4.1.2) obviously contains a misprint.} Vector Lyapunov functions are used to discusse controllability questions and to investigate decentralized control systems with feedback control. A result on optimal stabilization from the third author’s paper [Mathematical control theory, Banach Cent. Publ. 14, 383-400 (1985; Zbl 0582.93049)] is included. Results on set-valued differential inequalities, going back to K. Deimling, are stated and, again in combination with vector Lyapunov functions, are used to obtain results on the practical stability of a set-valued differential equation \(x'\in F(t,x)\), \(x\in\mathbb{R}^ n\), from the practical stability of an (also set-valued) comparison equation \(u'\in G(t,u)\), \(u\in\mathbb{R}^ N\) with \(N<n\).

The book consists of four chapters. Chapter 1 starts with definitions: a system (1) \(x'=f(t,x)\) where \(f\in C[\mathbb{R}_ +\times\mathbb{R}^ n,\mathbb{R}^ n]\) is said to be “practically stable if, given \((\lambda,A)\) with \(0<\lambda<A\), we have \(| x_ 0|<\lambda\) implies \(| x(t)|<A\), \(t\geq t_ 0\) for some \(t_ 0\in\mathbb{R}_ +\)”, where, as usual, \(x(t)\) denotes the solution of (1) with initial value \(x(t_ 0)=x_ 0\). {The reviewer has to confess that this definition seems to him to be somewhat fuzzy. What do the words “given \((\lambda,A)\)” really mean? Of course the decision whether or not a particular system is practically stable strongly depends on the chosen values of \(\lambda\) and \(A\), and so the notion, say, “\((\lambda,A)\)-practically stable” would be more appropriate.} In addition to the definition above, 11 modifications are given, culminating in the notion “eventually uniformly strongly practically stable”. After that, criteria for practical stability are presented, using scalar comparison systems (2) \(u'=g(t,u)\) where \(u\in C(\mathbb{R}_ +\times\mathbb{R}_ +,\mathbb{R})\), and stating conditions under which the practical stability of (2) implies practical stability of (1). Analogous criteria are also given for delay differential systems, integro-differential equations, difference equations, and impulsive differential equations.

In chapter 2, Lyapunov functions are introduced as additional tools for investigating practical stability. As usual, the Lyapunov functions are assumed to satisfy an inclusion \(a(| x|)\leq V(x,t)\leq b(| x|)\) with non-decreasing functions \(a(r)\), \(b(r)\); in addition, these functions now have to satisfy the inequality \(a(\lambda)<b(A)\). More general approaches using perturbing Lyapunov functions or vector Lyapunov functions are discussed. For large-scale systems a decomposition- aggregation method is presented. Then, inspired by the geometrical interpretation of the Lyapunov function level sets, more general definitions of practical stability in terms of arbitrary sets (i.e., \(x_ 0\in S_ 0(t_ 0)\) implies \(x(t)\in S(t)\) for \(t\geq t_ 0)\) or in terms of two measures (i.e., \(h_ 0(t_ 0,x_ 0)<\lambda\) implies \(h(t,x)<A\) for \(t\geq t_ 0)\) are introduced, and also with respect to these new notions stability criteria are obtained.

Chapter 3 is devoted to perturbed systems (3) \(x'=f(t,x)+R(t,x)\). The notion of practical stability under constantly acting perturbations (i.e., \(x_ 0\in S_ 0(t_ 0)\) and \(R(t,x)\in P\) for \(x\in S(t)\), \(t\geq t_ 0\) imply \(x(t)\in S(t)\) for \(t\geq t_ 0)\) and a new comparison theorem are presented, and again a large number of stability criteria, for systems (3) as well as for various other types of nonlinear systems, are obtained.

In chapter 4, control systems (4) \(x'=f(t,x,n)\) are considered. Using a Lyapunov function, conditions are formulated which assure that the practical stability properties of (4) are determined by the corresponding properties of a scalar comparison equation. {Note that this comparison equation should read \(w'=g(t,w,v)\). Equation (4.1.2) obviously contains a misprint.} Vector Lyapunov functions are used to discusse controllability questions and to investigate decentralized control systems with feedback control. A result on optimal stabilization from the third author’s paper [Mathematical control theory, Banach Cent. Publ. 14, 383-400 (1985; Zbl 0582.93049)] is included. Results on set-valued differential inequalities, going back to K. Deimling, are stated and, again in combination with vector Lyapunov functions, are used to obtain results on the practical stability of a set-valued differential equation \(x'\in F(t,x)\), \(x\in\mathbb{R}^ n\), from the practical stability of an (also set-valued) comparison equation \(u'\in G(t,u)\), \(u\in\mathbb{R}^ N\) with \(N<n\).

Reviewer: W.Müller (Berlin)

### MSC:

34D20 | Stability of solutions to ordinary differential equations |

34K20 | Stability theory of functional-differential equations |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

39A11 | Stability of difference equations (MSC2000) |

34A37 | Ordinary differential equations with impulses |

34A60 | Ordinary differential inclusions |

93D15 | Stabilization of systems by feedback |