This book expounds the theory of non-linear vibrations, a topic of great interest at present because of its many applications to important fields in physics and engineering. After introducing chapters giving the basic techniques for the study of non-linear systems the authors develop in detail the theory of selected topics encountered in their own work, presenting original material, approaches and results of analysis, and providing illustrations of useful applications.

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We show that diffusive couplings are necessary for minimization of cost functionals integrating quadratic synchronization error and quadratic input signals. This holds for identical linear systems with eigenvalues either on the imaginary axis or in the open left halfplane, whilst for eigenvalues in the open right half-plane, we present a counterexample in which the strong solution to the associated algebraic Riccati equation is not diffusive. For nonidentical systems satisfying the internal model principle for synchronization, we show that a certain part of the coupling must be diffusive. For equally chosen weights in the cost functional, we show that the dimension of the associated algebraic Riccati equation can be reduced significantly.We study an optimal control problem for coupled linear dynamical systems. In particular, we investigate the structure of the coupling that is optimal with respect to a cost functional that integrates quadratic synchronization error and quadratic input signals. We show that such a coupling is necessarily diffusive. Related Literature. Borrelli and Keviczky minimize a cost functional integrating quadratic synchronization error, quadratic input signals, and quadratic solutions, such that they arrive at a closed loop with asymptotically stable origin [1]. A Similar cost functional was proposed by Deshpande et al. [2]. Cao and Ren design couplings for integrators by minimizing a cost functional integrating quadratic synchronization error and quadratic input signals, whilst imposing the constraint that the coupling is diffusive [3]. Fardad et al. study sparse optimal couplings for the synchronization of oscillator networks and therein impose the constraint that the coupling is diffusive [4].In contrast to these works, we are concerned with optimal couplings for linear systems when minimizing a cost functional integrating quadratic synchronization error and quadratic input signals without integrating quadratic solutions and without imposing the constraint that the coupling is diffusive. As a consequence, the origin is not necessarily asymptotically stable in the closed loop, admitting for nontrivial solutions such as periodic orbits. Our main result is that the coupling which is optimal with respect to the considered cost functional is a diffusive coupling, even if we did not impose the constraint that the coupling should be diffusive. As a consequence, diffusive couplings are necessary for optimality when minimizing a cost functional of the considered form. Structure of the Paper. In section I, we formalize our problem. In section II, we answer the question whether diffusive couplings are necessary to minimize a cost functional which integrates quadratic synchronization error and quadratic input signals in the affirmative for systems with eigenvalues in the open left half-plane or on the imaginary axis. We present two corollaries of these results within section III that apply to nonidentical systems and to cost functionals with identical weights for every system, respectively. In the l...

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