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.
