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Singularities of plane into plane mappings described by parabolic two-component systems of quasi-linear partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney’s approach to such a case are discussed. Hierarchy of singularities is analyzed by the double-scaling expansion method for the simplest $2$-component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by $(k+1,k+2)$ curves which are of cusp type for $k=2n+1$, $n=1,2,3,\dots .$ Regularization of these singularities by deformation of plane into plane mappings into surface $S^{2+k}(\subset {ℝ}^{2+k})$ to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted. We finally compare the results obtained for the parabolic case with non-generic gradient catastrophes for hyperbolic systems.

This paper concerns control problems for multidimensional linear parabolic equations subject to hard/pointwise constraints on both boundary controls and state dynamic/output functions in the presence of uncertain perturbations within given regions. Such problems are formalized as minimax problems of optimal control, where the control strategy is sought as a feedback law depending on the current state position. Problems of this type are among the most important while the most challenging and difficult in control theory and applications. Based on the Maximum Principle in the theory of parabolic equations and on time convolutions in the theory of Fourier transforms, we reduce the problems under consideration to certain asymmetric games. This allows us to discover significant properties of feasible and optimal feedback controls for constrained parabolic systems.

The initial value problem , in all of the space, for the spatio-temporal FitzHugh-Nagumo equations is analyzed. When the reaction kinetics of the model can be outlined by means of piecewise linear approximations, then the solution of is explicitly obtained. For periodic initial data are possible damped travelling waves and their speed of propagation is evaluated. The results imply applications also to the non linear case.