13,241 research outputs found
Path-dependent Hamilton-Jacobi equations in infinite dimensions
We propose notions of minimax and viscosity solutions for a class of fully
nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators
on Hilbert space. Our main result is well-posedness (existence, uniqueness, and
stability) for minimax solutions. A particular novelty is a suitable
combination of minimax and viscosity solution techniques in the proof of the
comparison principle. One of the main difficulties, the lack of compactness in
infinite-dimensional Hilbert spaces, is circumvented by working with suitable
compact subsets of our path space. As an application, our theory makes it
possible to employ the dynamic programming approach to study optimal control
problems for a fairly general class of (delay) evolution equations in the
variational framework. Furthermore, differential games associated to such
evolution equations can be investigated following the Krasovskii-Subbotin
approach similarly as in finite dimensions.Comment: Final version, 53 pages, to appear in Journal of Functional Analysi
Minimax frequency domain performance and robustness optimization of linear feedback systems
It is shown that feedback system design objectives, such as disturbance attenuation and rejection, power and bandwidth limitation, and robustness, may be expressed in terms of required bounds of the sensitivity function and its complement on the imaginary axis. This leads to a minimax frequency domain optimization problem, whose solution is reduced to the solution of a polynomial equation
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