DOI: 10.68381/jca34035 ISSN: 0944-6532
Excess-Continuous Prox-Regular Sweeping Processes
Vincenzo Recupero, Federico Stra
We consider the Moreau's sweeping processes driven by a time dependent prox-regular set
\mathcal{C}(t)
C
(
t
)
which is continuous in time with respect to the asymmetric distance
e
e
called the excess, defined by
e\,(\mathcal{A},\mathcal{B}) := \sup_{x \in \mathcal{A}} d(x,\mathcal{B})
e
(
A
,
B
)
:
=
sup
x
∈
A
d
(
x
,
B
)
for every pair of sets
\mathcal{A}
A
,
\mathcal{B}
B
in a Hilbert space. As observed by J.J. Moreau in his pioneering works, the excess provides the natural topological framework for sweeping processes. Assuming a uniform interior cone condition for
\mathcal{C}(t)
C
(
t
)
, we prove that the associated sweeping process has a unique solution, thereby improving the existing result on continuous prox-regular sweeping processes in two directions: indeed, in the previous literature
\mathcal{C}(t)
C
(
t
)
was supposed to be continuous in time with respect to the symmetric Hausdorff distance instead of the excess and also its boundary
\partial\mathcal{C}(t)
∂
C
(
t
)
was required to be continuous in time, an assumption which we completely drop. Therefore our result allows to consider a much wider class of continuously moving constraints.