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On a new superposition principle for optimization problems. (English) Zbl 0595.49020

Sémin., Équations Dériv. Partielles 1985-1986, Exposé No. 24, 14 p. (1986).
The superposition principle in quantum mechanics and in wave optics is an important primary physical principle. However, it yields a fact which is very simple in mathematical sense, i.e. linearity of equations in quantum mechanics and wave optics. We show, that some new superposition principles used to solve some nonlinear equations including the Bellman and Hamilton-Jacobi equations lead to ”linearity” of these equations in some semi-modules, what allows to apply some formulas and results of usual linear equations to this case.
Hopf has constructed a solution of a quasi-linear equation, starting from Burgers equation with small viscosity and then passing to the limit at the viscosity tends to zero. He used the fact, that Burgers equation may be reduced to the heat equation by a certain substitution.
We consider this situation from a different viewpoint, using an integrated variant of Burgers equation.
We show the main ideas using simple examples and omitting proofs.

MSC:

49L99 Hamilton-Jacobi theories
35Q99 Partial differential equations of mathematical physics and other areas of application
49S05 Variational principles of physics
35L60 First-order nonlinear hyperbolic equations
49L20 Dynamic programming in optimal control and differential games
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