arXiv subjects
Publications and source records attributed to Sławomir Plaskacz.
In this paper, for the Hamilton-Jacobi-Bellman equation with an infinite horizon and state constraints, we construct a suitably regular representation. This allows us to reduce the problem of existence and uniqueness of solutions to the Frankowska and Basco theorem from (2019). Furthermore, we demonstrate that our representations are stable. The obtained results are illustrated with examples.
Single species fisheries and prey-predator models with marine protected areas (MPA) are studied. The single species case is considered when the fishing effort is around the species extinction threshold and the influence of implementing MPA on catch quantity are studied. In the prey-predator fishery model, the situation with the fishing effort close to the predator extinction value is considered and the effects of implementing MPA are discussed with the focus on MPA sizes assuring an acceptable catch level (i.e. food security) and the sustainability of both the predator and prey populations.
We describe the family of initial conditions for Hamilton-Jacobi equations (HJE) corresponding to optimal control problems that can be retrieved by solving HJE backward in time.
The paper investigates sufficient conditions on a differential inclusion which guarantee that the origin is a finite time stable equilibrium, namely a weak local one, a weak global one or a strong local one. The analysis relies on the existence of a Lyapunov function. A new Gronwall type results are used to estimate the settling time. An example of a neural network which is finite-time stable is given
We consider a 3-player game in the normal form, in which each player has two actions. We assume that the game is symmetric and repeated infinitely many times. At each stage players make their choices knowing only the average payoffs from previous stages of all the players. A strategy of a player in the repeated game is a function defined on the convex hull of the set of payoffs. Our aim is to construct a strong Nash equilibrium in the repeated game, i.e. a strategy profile being resistant to deviations by coalitions. Constructed equilibrium strategies are safe, i.e. the non-deviating player payoff is not smaller than the equilibrium payoff in the stage game, and deviating players' payoffs do not exceed the non-deviating player payoff more than a positive constant which can be arbitrary small and chosen by the non-deviating player. Our construction is inspired by Smale's good strategies described in \cite{smale}, where the repeated Prisoner's Dilemma was considered. In proofs we use arguments based on approachability and strong approachability type results.
We consider the repeated prisoner's dilemma (PD). We assume that players make their choices knowing only average payoffs from the previous stages. A player's strategy is a function from the convex hull $\mathfrak{S}$ of the set of payoffs into the set $\{C,\,D\}$ ($C$ means cooperation, $D$ -- defection). S. Smale in \cite{smale} presented an idea of good strategies in the repeated PD. If both players play good strategies then the average payoffs tends to the payoff corresponding to the profile $(C,C)$ in PD. We adopt the Smale idea to define semi-cooperative strategies - players do not take as a referencing point the payoff corresponding to the profile $(C,C)$, but they can take an arbitrary payoff belonging to the $β$-core of PD. We show that if both players choose the same point in the $β$-core then the strategy profile is an equilibrium in the repeated game. If the players choose different points in the $β$-core then the sequence of the average payoffs tends to a point in $\mathfrak{S}$. The obtained limit can be treated as a payoff in a new game. In this game the set of players' actions is the set of points in $S$ that corresponds to the $β$-core payoffs.