Liapunov functions and boundedness

In this paper we give conditions under which one can conclude that all solutions of a differential equation are bounded when there is a Liapunov function which is not radially unbounded. The problem of Lurie and the Liénard equation are given as examples.     

Overlapping block diagonal dominance and existence of Liapunov functions

The main objective of this paper is to formulate a generalization of block diagonal dominance, which can be used to establish nonsingularity of matrices via overlapping diagonal blocks. A number of stability results are derived in the new setting by exploiting the well-known M-matrix properties, as well as extensions of the normalization, scaling, and alternative norm utilization. A link between generalized block diagonal dominance and vector Liapunov functions is established, which can be applied in the stability analysis of interconnected dynamic systems.     

Hierarchical Liapunov functions

A complex system is considered as a collection of dynamic components which may be combined in different ways to form stable subassemblies (subsystems), which may serve again as units to build still more complex stable configurations. A hierarchical Liapunov function is proposed, which can be used to determine stability of the overall system obtained by the multilevel interconnection process. Stability when established in this way is hierarchically connective. That is, if at any instant in time, the hierarchical evolution of the system is interrupted, the system can fall apart in exactly the same way it was constructed and, by re-starting the process, put together again without loss of stability. This fact may be used to explain the evolution of stable forms in natural as well as man-made systems.     

Liapunov functions and error bounds for approximate solutions of ordinary differential equations

Summary It is shown that Liapunov functions may be used to obtain error bounds for approximate solutions of systems of ordinary differential equations. These error bounds may reflect the behaviour of the error more accurately than other bounds.     

Contours of Liapunov functions

As is well known, the stability of a dynamical system in two dimensions may be demonstrated in a very intuitive fashion from the existence of a suitable positive-definite Liapunov function, providing the contours of this function in a neighborhood of the stable point are Jordan curves. It is shown that the Liapunov function will certainly have this property if the stable point is an isolated stationary point in the sense of the Clarke calculus, but a counterexample is given if this assumption is weakened to the stable point being an isolated local extremum.This work was carried out with the support of the Natural Sciences and Engineering Council of Canada, which is gratefully acknowledged.     

Almost global existence of solutions of quasi-linear hyperbolic equations

We consider the quasi-linear hyperbolic initial value problem (1) of the Introduction, and prove that for any T>0 there is a bound such that if the norm of the initial data is smaller than that bound then the solution of (1) exists on all of [0, T].     

On global existence of solutions to a cross-diffusion system

We consider a strongly coupled nonlinear parabolic system which arises in population dynamics in n-dimensional domains (n?1). We prove the global existence of classical solutions to the system for n<10.     

Lyapunov vector functions and partial boundedness of solutions with partially controlled initial conditions

On the basis of the method of Lyapunov vector functions, we obtain a sufficient test for the uniform partial boundedness of solutions with partially controlled initial conditions. We introduce the notions of partial equiboundedness, partial equiboundedness in the limit, and partial uniform boundedness in the limit of solutions with partially controlled initial conditions. By the method of Lyapunov vector functions, we obtain sufficient tests for the partial equiboundedness of solutions and for the partial uniform boundedness in the limit and partial equiboundedness in the limit of solutions with partially controlled initial conditions.     

Chemotaxis with logistic source: Very weak global solutions and their boundedness properties

We consider the chemotaxis system

     

Global existence in the future and boundedness of submanifolds of solutions of a scalar comparison equation

Sufficient conditions for the global existence in the future, and for the boundedness, of solutions of the scalar differential comparison equation $\dot u = a\left( t \right)f\left( u \right) + b\left( t \right)g\left( u \right)$ are provided.     

Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model

Using the energy estimate and Gagliardo–Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for a strongly coupled reaction–diffusion system are proved. This system is the Shigesada–Kawasaki–Teramoto three-species cooperating model with self- and cross-population pressure. Meanwhile, some criteria on the global asymptotic stability of the positive equilibrium point for the model are also given by Lyapunov function. As a by-product, we proved that only constant steady states exist if the diffusion coefficients are large enough.