Dan Marchesin

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The purpose of this paper is to classify the solutions of Riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws. Hyperbolic singularities play the role in the theory of Riemann problems that rest points play in the theory of ordinary differential equations: Indeed, generically, only a finite number of structures can(More)
We determine the bifurcation from the constant solution of nonclassical transitional and overcompressive viscous shock prooles, in regions of strict hyperbolicity. Whereas classical shock waves in systems of conservation laws involve a single characteristic eld, nonclassical waves involve two elds in an essential way. This feature is reeected in the viscous(More)
The Schatten p-norm condition of the discrete-time Lyapunov operator L A defined on matrices P 2 R nn by L A P P ? APA T is studied for stable matrices A 2 R nn. Bounds are obtained for the norm of L A and its inverse that depend on the spectrum, singular values and radius of stability of A. Since the solution P of the the discrete-time algebraic Lyapunov(More)
New upper bounds for the solution of the discrete algebraic Lyapunov equation (DALE) P = APA T + Q are presented. The only restriction on their applicability is that A be stable; there are no restrictions on the singular values of A nor on the diagonalizability of A. The new bounds relate the size of P to the radius of stability of A. The upper bounds are(More)
A family of sharp, arbitrarily tight, upper and lower matrix bounds for solutions of the discrete algebraic Lyapunov are presented. The lower bounds are tighter than previously known ones. Unlike the majority of previously known upper bounds, those derived here have no restrictions on their applicability. Upper and lower bounds for individual eigenvalues(More)
We consider a one-parameter family of nonstrictly hyperbolic systems of conservation laws modeling three-phase flow in a porous medium. For a particular value of the parameter, the model has a shock wave solution that undergoes several bifurcations upon perturbation of its left and right states and the parameter. In this paper we use singularity theory and(More)