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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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Generalized Characteristics of First Order PDEs - Applications in Optimal Control and Differential Games (Hardcover, 1998 ed.)
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Generalized Characteristics of First Order PDEs - Applications in Optimal Control and Differential Games (Hardcover, 1998 ed.)
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In some domains of mechanics, physics and control theory boundary
value problems arise for nonlinear first order PDEs. A well-known
classical result states a sufficiency condition for local existence
and uniqueness of twice differentiable solution. This result is
based on the method of characteristics (MC). Very often, and as a
rule in control theory, the continuous nonsmooth
(non-differentiable) functions have to be treated as a solutions to
the PDE. At the points of smoothness such solutions satisfy the
equation in classical sense. But if a function satisfies this
condition only, with no requirements at the points of
nonsmoothness, the PDE may have nonunique solutions. The uniqueness
takes place if an appropriate matching principle for smooth
solution branches defined in neighboring domains is applied or, in
other words, the notion of generalized solution is considered. In
each field an appropriate matching principle are used. In Optimal
Control and Differential Games this principle is the optimality of
the cost function. In physics and mechanics certain laws must be
fulfilled for correct matching. A purely mathematical approach also
can be used, when the generalized solution is introduced to obtain
the existence and uniqueness of the solution, without being aimed
to describe (to model) some particular physical phenomenon. Some
formulations of the generalized solution may meet the modelling of
a given phenomenon, the others may not.
General
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