Figure 4 - uploaded by Maurizio Falcone
Content may be subject to copyright.
Source publication
In this paper we present some numerical methods for the solution of two-persons zero-sum deterministic differential games. The methods are based on the dynamic programming approach. We first solve the Isaacs equation associated to the game to get an approximate value function and then we use it to reconstruct approximate optimal feedback controls a...
Context in source publication
Context 1
... the numerical experiment below we have chosen v P = 2, v E = 1, A = [ 3 4 π, 3 4 π] and B = [−π, π]. As one can see in Figure 4 the time of capture at points which are below the origin and which cannot be reached by P in a direct way have a value bigger than at the symmetric points (above the origin). This is clearly due to the fact that P has to zig-zag to those points because the directions pointing directly to them are not allowed ( Figure 5). ...
Similar publications
For the particularity of warfare hybrid dynamic process, a class of warfare
hybrid dynamic systems is established based on Lanchester equation in a battle, where a heterogeneous force of different troop types faces a homogeneous force. This model can be characterized by the interaction of continuous-time models (governed by Lanchester equation), an...
Differential games are a combination of game theory and optimum control methods. Their solutions are based on Bellman's principle of optimality. In this paper, the zero-sum differential game theory has been used for the purposes of controlling a mechatronic object: a single-link manipulator. In this case, analytical solutions are unavailable, thus...
This paper addresses the problem of utility maximization under uncertain parameters. In contrast with the classical approach, where the parameters of the model evolve freely within
a given range, we constrain them via a penalty function. We show that this robust optimization process can be interpreted as a two-player zero-sum stochastic differentia...
This paper analyzes dynamic advertising and pricing policies in a durable-good duopoly. The proposed infinite-horizon model, while general enough to capture dynamic price and advertising interactions in a competitive setting, also permits closed-form solutions. We use differential game theory to analyze two different demand specifications – linear...
In this paper, we investigate the differential game theoretic approach for distributed dynamic cooperative power control in cognitive radio ad hoc networks (CRANETs). First, a payoff function is defined by taking into consideration the tradeoff between the stock of accumulated power interference to the primary networks and the dynamic regulation of...
Citations
... Then a maneuver is assumed to be the defined group of airplane conditions to be executed during a period of time; thus, a change or remaining on any condition at any time defines the beginning and ending of a maneuver [10][11][12][13][14][15][18][19][20][24][25][26][27][28][29][30]. Nevertheless, there is a minimum of information required that defines the original aircraft status at the beginning of the simulation [24][25][26][34][35][36][37][38][39][40][41]. ...
... The first step in developing a flight simulator is to create a virtual environment on the computer screen where the user can appreciate the aircraft path [1][2][3][4][7][8][9][10][34][35][36][37][38][39][40][41]. The aircraft movement must be represented appropriately, and physical and mathematical equations must be solved using the programming of numerical methods. ...
... Time, aircraft speed, and angular positions during every maneuver are used as initial and final conditions to create a simulation of the aircraft's animation, and the resulted information is used as input to create an animation of the movement as would be in the aeronautical instruments [29][30][31][37][38][39][40][41]. Then, in this work, the main procedure is a calculation routine that contains the equations for calculating the aircraft displacement along the 3D position. ...
This manuscript describes the computational process to calculate an airplane path and display it in a 2D and 3D coordinate system on a computer screen. The airplane movement is calculated as a function of its dynamic’s conditions according to physical and logical theory. Here, the flight is divided into maneuvers and the aircraft conditions are defined as boundary conditions. Then the aircraft position is calculated using nested loops, which execute the calculation procedure at every step time (Δt). The calculation of the aircraft displacement is obtained as a function of the aircraft speed and heading angles. The simulator was created using the C++ programming language, and each part of the algorithm was compiled independently to reduce the source code, allow easy modification, and improve the programming efficiency. Aerial navigation involves very complex phenomena to be considered for an appropriate representation; moreover, in this manuscript, the influence of the mathematical approach to properly represent the aircraft flight is described in detail. The flight simulator was successfully tested by simulating some basic theoretical flights with different maneuvers, which include stationary position, running along the way, take off, and some movements in the airspace. The maximum aircraft speed tested was 120 km/h, the maximum maneuver time was 12 min, and the space for simulation was assumed to be without obstacles. Here, the geometrical description of path and speed is analyzed according to the symmetric and asymmetric results. Finally, an analysis was conducted to evaluate the approach of the numerical methods used; after that, it was possible to confirm that precision increased as the step time was reduced. According to this analysis, no more than 500 steps are required for a good approach in the calculation of the aircraft displacement.
... When the game dynamics and cost functions are known, the saddle-point equilibrium of a two-player zero-sum SDG can be characterized by the Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE). The analytical solutions of the HJI PDEs are in general not available, and one needs to resort on numerical methods such as grid-based approaches [4], [5] to solve these PDEs approximately. However, the grid-based approaches suffer from curse of dimensionality, making them computationally intractable for systems with large dimensions [6]. ...
This paper addresses a continuous-time risk-minimizing two-player zero-sum stochastic differential game (SDG), in which each player aims to minimize its probability of failure. Failure occurs in the event when the state of the game enters into predefined undesirable domains, and one player's failure is the other's success. We derive a sufficient condition for this game to have a saddle-point equilibrium and show that it can be solved via a Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE) with Dirichlet boundary condition. Under certain assumptions on the system dynamics and cost function, we establish the existence and uniqueness of the saddle-point of the game. We provide explicit expressions for the saddle-point policies which can be numerically evaluated using path integral control. This allows us to solve the game online via Monte Carlo sampling of system trajectories. We implement our control synthesis framework on two classes of risk-minimizing zero-sum SDGs: a disturbance attenuation problem and a pursuit-evasion game. Simulation studies are presented to validate the proposed control synthesis framework.
... Understanding the set of discontinuities is of major importance when implementing numerical methods to approximate the discontinuous viscosity solution to the HJI equation. Indeed, most numerical schemes provide provable good approximations of the value only in compact sets away from the discontinuities [1,4,19]. ...
... There are several numerical methods that can be used to approximate the value of the game by solving a discrete approximation of the above boundary-value problem. The semi-Lagrangian schemes [1,4,19] are iterative methods based on finite differences that make use of the dynamic programming principle to compute the value of a discrete version of the game defined on a grid. These schemes are proven to converge to the viscosity solution in sets where this one is continuous, even though it can be also implemented in the presence of discontinuities. ...
We consider a surveillance-evasion game in an environment with obstacles. In such an environment, a mobile pursuer seeks to maintain the visibility with a mobile evader, who tries to get occluded from the pursuer in the shortest time possible. In this two-player zero-sum game setting, we study the discontinuities of the value of the game near the boundary of the target set (the non-visibility region). In particular, we describe the transition between the usable part of the boundary of the target (where the value vanishes) and the non-usable part (where the value is positive). We show that the value enjoys a different behaviour depending on the regularity of the obstacles involved in the game. Namely, we prove that the boundary profile is continuous for the case of smooth obstacles, and that it exhibits a jump discontinuity when the obstacle contains corners. Moreover, we prove that, in the latter case, there is a semi-permeable barrier emanating from the interface between the usable and the non-usable part of the boundary of the target set.
... In general, it might not be easy to find explicit solutions for differential game problems, even if we restrict ourselves to a small amount of players, and numerical methods have to be employed (see [13] or [15]). If collocation methods are employed, as the number of players increases, we have to deal with the so called "curse of dimensionality", which might boost the computational cost of the numerical methods. ...
This paper concerns the design of a multidimensional Chebyshev interpolation based method for a differential game theory problem. In continuous game theory problems, it might be difficult to find analytical solutions, so numerical methods have to be applied. As the number of players grows, this may increase computational costs due to the curse of dimensionality. To handle this, several techniques may be applied and paralellization can be employed to reduce the computational time cost. Chebyshev multidimensional interpolation allows efficient multiple evaluations simultaneously along several dimensions, so this can be employed to design a tensorial method which performs many computations at the same time. This method can also be adapted to handle parallel computation and, the combination of these techniques, greatly reduces the total computational time cost. We show how this technique can be applied in a pollution differential game. Numerical results, including error behaviour and computational time cost, comparing this technique with a spline-parallelized method are also included.
... While it can be applied to a general class on -player SDG, we present the algorithm adapted to the SDG model for the couple's problem described in the preceding section. The SDG model is discretized in a Semi-Lagrangian way (see, for instance, 32 ). This implies first discretizing in time and space and then using numerical interpolation, by means of radial basis functions (see 19 ). ...
... The Dynamic Programming Principle in discrete time implies that the discrete value functions ℎ satisfy (see 32 ) ...
We introduce an algorithm to find feedback Nash equilibria of a stochasticdifferential game. Our computational approach is applied to analyze optimalpolicies to nurture a romantic relationship in the long term. This is a fundamen-tal problem for the applied sciences, which is naturally formulated in this workas a stochastic differential game with nonlinearities. We use our computationalmodel to analyze the risk of marital breakdown. In particular, we introduce theconcept of “love at risk,” which allows us to estimate the probability of a couplebreaking up in the face of possible unfavorable scenarios.
... Researchers have considered many problems (cf., Bonanno, 2013;Chen, 2021;Falcone, 2006;Gromova et al., 2016;Li, 2014;Lopez, 2020;Mazalov, 2014;Mazalov et al., 2018;Sedakov et al., 2013) simultaneously in crisp and fuzzy environments corresponding to game theory. Numerous articles have been published involving different game theory strategies to cover various real-life problems, such as online-shopping marketing management problem (Bhaumik et al., 2017), water resources management problem (Roy & Bhaumik, 2018), tourism management problem (Bhaumik et al., 2021), and pollution control (Gromova et al., 2016). ...
Strategy considerations for optimal outcomes play the key roles in every decision-making problems. Game theory in itself preserves the sustainability to draw some decision-making conclusions. Different forms of game exist in literature. This article is concerned with two-player nonlinear static game. We consider here the solution of the game problems through the algorithmic structures, with some modification, of Hooke–Jeeves (HJ) algorithm which is a tool to encounter unconstrained optimization. Basically, we lay emphasis on the modification of the considered function which is to be optimized. Hence, the name of the proposed algorithm is modified HJ algorithm. Numerous articles present different solutions of various real-life problems using HJ algorithm. Forest management problem is considered here under modified HJ algorithm to get solution. This article concludes with some comparative analysis with respect to some different methods, for example, analytical method and graphical method. Our proposed methodology gives better results than the results obtained in analytical method and graphical method with respect to optimal strategy sets and optimal payoff values.
... Although we present here a discretization and computational implementation of the model introduced above, it can be easily extended to N -players with more than one control and state variables per player. The presented model is discretized in a Semi-Lagrangian way (see, for instance, [5]). Thus, the discrete version of (1.2) is ...
We introduce a new algorithm to find feedback Nash
equilibria of stochastic differential games and, secondly, we apply our methodology to a problem
of significance in the social sciences, related to human behavior.
... The model is discretized in a Semi-Lagrangian way (see, for instance, 27 ). This implies first discretizating in time and space and the using numerical interpolation (in this case, using radial base functions-see 14 ). ...
... The Dynamic Programming Principle (DPP) in discrete time implies that the discrete value functions satisfy (see 27 ) ...
We introduce a new algorithm to find feedback Nash equilibria of a stochastic differential game. Our computational approach is applied to analyze optimal policies to nurture a romantic relationship in the long term. This is a fundamental problem for the applied sciences, which is naturally formulated in this work as a stochastic differential game with nonlinearities. We use our computational model to analyze the risk of marital breakdown. In particular, we introduce the concept of "love at risk" which allows us to estimate the probability of a couple breaking up in the face of possible unfavorable scenarios.
... The 300 solution was obtained for the model size down to 300 m depth, summing up to 1,200 grid points. These numerical parameters satisfy the von Neumann stability condition for explicit-in-time numerical scheme (e.g., Ames, 1977) for the material properties of ...
Deglaciation in Svalbard was followed-up by seawater ingression and the deposition of marine (deltaic) sediments in fjord valleys, while elastic rebound resulted in fast land uplift and the exposure of these sediment to the atmosphere, therefore the formation of epigenetic permafrost. This was then followed by the accumulation of aeolian sediments, which froze syngenetically. The permafrost was drilled in the east Adventdalen valley, Svalbard, 3–4 km from the maximum up-valley reach of post-deglaciation seawater ingression, and its ground ice was measured for chemistry. While ground ice in the syngenetic part is basically fresh the epigenetic part reveals a frozen fresh-saline water interface (FSI), with chloride concentrations increasing from the top of the epigenetic part (depth of 5.5 m) to about 15 % that of seawater at 11 m. We applied a one-dimensional freezing model in order to examine the rate of top-down permafrost aggradation, which could accommodate with the observed frozen FSI. The model examined permafrost development under different scenarios of mean average air temperature, water-freezing temperature and the degree of pore-water freezing. We found that even at the relatively high temperatures of the Early to mid-Holocene, permafrost could aggrade quite fast, e.g. down to 15 to 33 m in 200 years, therefore allowing freezing of the fresh-saline water interface despite of the relatively fast rebound rate and the resultant increase in topographic gradients toward the sea. This could be aided by non-complete pore water freezing, which possibly lead to slightly faster aggradation, resulting in the freezing of the entire marine section at that location (23 m) within less than 200 years. We conclude that freezing should have occurred immediately after the exposure of the marine sediment to atmospheric conditions.
... Deterministic two-player zero-sum games lead to first-order Isaacs equations. We refer for the numerical approximation of deterministic problems to [5,7,11,12] and the references therein. An important field of application is H ∞ control, for which we highlight [3,13,22,29]. ...
We study monotone P1 finite element methods on unstructured meshes for fully non-linear, degenerately parabolic Isaacs equations with isotropic diffusions arising from stochastic game theory and optimal control and show uniform convergence to the viscosity solution. Elliptic projections are used to manage singular behaviour at the boundary and to treat a violation of the consistency conditions from the framework by Barles and Souganidis by the numerical operators. Boundary conditions may be imposed in the viscosity or in the strong sense, or in a combination thereof. The presented monotone numerical method has well-posed finite dimensional systems, which can be solved efficiently with Howard’s method.
