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## Stochastic Optimal Control (2014)

### Citations

5173 | Probability and Measure - Billingsley - 1995 |

2502 | Robot Motion Planning - Latombe - 1991 |

1342 | Real-time obstacle avoidance for manipulators and mobile robots,”
- Khatib
- 1986
(Show Context)
Citation Context ...6] provides resolution completeness, which means a valid solution, if one exists, is returned when the resolution parameters are set fine enough. The second practical approach called potential fields =-=[67]-=- provides complete solutions by using appropriate navigation functions. Although the two approaches can be applied to problems with state spaces of up-to five dimensions, cell decomposition methods su... |

1321 |
Optimum consumption and portfolio rules i a continuous time model
- Merton
- 1971
(Show Context)
Citation Context ...d are known to be computationally challenging (see, e.g., [27--31]). Problems with closed-form solutions such as the linear quadratic gaussian (LQG) problem [17, 32, 33] or Merton's portfolio problem =-=[34]-=- are rare. These exceptions are due to special problem structures such as a quadratic value function for the LQG problem or an optimal constant-fraction investment strategy for Merton's portfolio prob... |

1273 | Probabilistic Roadmaps for Path Planning in High-Dimensional Configuration Spaces
- Kavraki, Svestka, et al.
- 1996
(Show Context)
Citation Context ...rge number of obstacles. Sampling-based algorithms Therefore, to overcome the above difficulties, a class of sampling-based algorithms for the motion planing problem have been studied since the 1990s =-=[50, 70-75]-=-. The main advantage of these algorithms is to avoid such an explicit representation of obstacles in the configuration space by using feasibility tests of candidate trajectories. This leads to signifi... |

993 |
Tsitsiklis, Parallel and Distributed Computation: Numerical Methods
- Bertsekas, N
- 1989
(Show Context)
Citation Context ...thms such as policy iteration, value iteration and their parallel versions to find e-optimal solutions. The thorough treatment of these algorithms can be found in the work by Bertsekas and Tsitsiklis =-=[41,42]-=-. However, having a single MDP to approximate the continuous problem often looses the fidelity of the original continuous problem model. Furthermore, assigning ad-hoc transition probabilities on the M... |

681 | Recursive Methods in Economic Dynamics - Stokey, Lucas - 1989 |

625 | Randomized kinodynamic planning,”
- LaValle, Jr
- 2001
(Show Context)
Citation Context ...nds to grow towards large unsearched areas. Several variants of the RRT algorithm have been studied extensively [78,85-92] and shown to work very well for systems with nonlinear differential dynamics =-=[71,78]-=-. The algorithm has also been implemented on several robotic platforms [10, 93-96]. We emphasize that besides avoiding an explicit representation of obstacles in the configuration space, the RRT algor... |

597 |
The Complexity of Robot Motion Planning
- Canny
- 1988
(Show Context)
Citation Context ...n space by Reif in 1979 [611. In addition, in 1988, Canny showed that computing the shortest path in a three-dimensional Euclidean space populated with obstacles is NP-hard in the number of obstacles =-=[62]-=-. Therefore, the optimal motion planning is computationally challenging even when the dimension of the configuration space is fixed. As the optimal motion planning problem can be thought as a"stochast... |

550 |
Viscosity solutions of Hamilton–Jacobi equations
- Crandall, Lions
- 1983
(Show Context)
Citation Context ...ave focused on new solution concepts that allow for non-smooth value functions. Since 1983, viscosity solutions have gained popularity as an alternative and natural solution concept for HJB equations =-=[17, 20]-=-. Viscosity solutions are a weak formulation of solutions to HJB equations that enables us to continue the dynamic programming approach. For a large class of optimal control problems, the value functi... |

543 | Random Geometric Graphs - Penrose - 2003 |

538 | Critical power for asymptotic connectivity in wireless networks - Gupta, PR - 1998 |

487 |
Recursive Macroeconomic Theory
- Ljungqvist, Sargent
- 2000
(Show Context)
Citation Context ...imal control are often derived for engineering systems such as mechanical systems. Due to their abstraction, these models are also applied for problems in other domains such as mathematical economics =-=[1, 2]-=- and mathematical finance [3]. Therefore, stochastic optimal control has been studied extensively by several research communities, and each community focuses on different theoretical and implementatio... |

456 | Deterministic and stochastic optimal control - Fleming, Rishel - 1975 |

442 | Principles of Robot Motion: Theory, Algorithms, and Implementations
- Choset, Lynch, et al.
- 2005
(Show Context)
Citation Context ... planning problem is to find a sequence of feasible control inputs to drive the system from the initial state to one of the goal states and at the same time avoid collision with obstacles (see, e.g., =-=[52, 59, 60]-=- and references therein). The optimal version of the problem called optimal motion planning seeks for a feasible solution that minimizes some performance measure. These problems, which can be cast as ... |

425 |
Controlled Markov Processes and Viscosity Solutions
- Fleming, Soner
- 1993
(Show Context)
Citation Context ...ave focused on new solution concepts that allow for non-smooth value functions. Since 1983, viscosity solutions have gained popularity as an alternative and natural solution concept for HJB equations =-=[17, 20]-=-. Viscosity solutions are a weak formulation of solutions to HJB equations that enables us to continue the dynamic programming approach. For a large class of optimal control problems, the value functi... |

404 | An algorithm for planning collision-free paths among polyhedral obstacles - Lozano-PCrez, Wesley - 1979 |

397 | Rapidly-exploring random trees: A new tool for path planning
- LaValle
- 1998
(Show Context)
Citation Context ... algorithm called the Rapidly-Exploring Random Tree (RRT) were proposed by LaValle and Kuffner to avoid the need to specify a priori samples and tailored for single-query motion planning applications =-=[51,84,85]-=-. The RRT algorithm constructs a tree-based structure connecting an initial state to a goal region, which efficiently searches non-convex high dimensional search spaces. The algorithm is designed to d... |

395 | The complexity of Markov decision processes - Papadimitriou, Tsitsiklis - 1987 |

341 |
Optimal control theory: an introduction
- Kirk
- 2004
(Show Context)
Citation Context ...osed-form or exact algorithmic solutions and are known to be computationally challenging (see, e.g., [27--31]). Problems with closed-form solutions such as the linear quadratic gaussian (LQG) problem =-=[17, 32, 33]-=- or Merton's portfolio problem [34] are rare. These exceptions are due to special problem structures such as a quadratic value function for the LQG problem or an optimal constant-fraction investment s... |

281 | Potential Field Methods and Their Inherent Limitations for Mobile Robot Navigation
- Koren, Borenstein
- 1991
(Show Context)
Citation Context ...imensions, cell decomposition methods suffer from the curse of dimensionality due to the large number of cells and difficult cell management [68], and potential field methods suffer from local minima =-=[69]-=-. More importantly, all previously mentioned methods require an explicit representation of the obstacles in the configuration space for the construction of solutions. Hence, these methods are not suit... |

227 | On the piano movers problem: II. General techniques for computing topological properties of real algebraic manifolds. - Schwartz, Sharir - 1983 |

225 | Real-time motion planning for agile autonomous vehicles,” in
- Frazzoli, Dahleh, et al.
- 2001
(Show Context)
Citation Context ... chooses an expanding node that is proportional to the size of its Voronoi region and tends to grow towards large unsearched areas. Several variants of the RRT algorithm have been studied extensively =-=[78,85-92]-=- and shown to work very well for systems with nonlinear differential dynamics [71,78]. The algorithm has also been implemented on several robotic platforms [10, 93-96]. We emphasize that besides avoid... |

218 | New lower bound techniques for robot motion planning problems - Canny, Reif - 1987 |

187 | Tsitsiklis. A survey of computational complexity results in systems and control
- Blondel, N
(Show Context)
Citation Context ...e LQG problem or an optimal constant-fraction investment strategy for Merton's portfolio problem. General continuous time and space problems can be solved approximately by discretizing time and space =-=[27]-=-. This discretization is used in numerical methods that solve H.JB equations or in the construction of approximating discrete-time finite-state Markov Decision Processes (MDPs). Discrete-time finite-s... |

186 |
Random plane networks
- Gilbert
- 1961
(Show Context)
Citation Context ...ns to connect edges, we have different random graph models. For instance, when an edge is formed if the distance between the two points is bounded by a positive constant, we have Gilbert's disc model =-=[101]-=-. Another popular model called k-nearest neighbor graph considers edges between k nearest neighbors [103]. A remarkable result in this field certifies that when k = O(log n) where n is the number of p... |

185 | Sampling-based algorithms for optimal motion planning
- Karaman, Frazzoli
(Show Context)
Citation Context ... planning problem is to find a sequence of feasible control inputs to drive the system from the initial state to one of the goal states and at the same time avoid collision with obstacles (see, e.g., =-=[52, 59, 60]-=- and references therein). The optimal version of the problem called optimal motion planning seeks for a feasible solution that minimizes some performance measure. These problems, which can be cast as ... |

162 | On the complexity of solving Markov decision problems - Littman, Dean, et al. - 1995 |

143 |
A subdivision algorithm in configuration space for findpath with rotation,”
- Brooks, Lozano-Perez
- 1985
(Show Context)
Citation Context ...on planning, which return a valid solution in finite time, if one exists, and failure otherwise, unsuitable for practical usage [63-65]. The first practical approach called cell decomposition methods =-=[66]-=- provides resolution completeness, which means a valid solution, if one exists, is returned when the resolution parameters are set fine enough. The second practical approach called potential fields [6... |

133 | Real-time randomized path planning for robot navigation,” - Bruce, Veloso - 2002 |

128 | Variable Resolution Discretization in Optimal Control
- Munos, Moore
- 2002
(Show Context)
Citation Context ...s and regularity theory for this class of PDEs are not well understood [38]. Thus, other methods approximate a continuous problem with a single discretetime finite-state Markov Decision Process (MDP) =-=[39,40]-=- without invoking the associated HJB equation. When dealing with finite-state MDPs, we can use specialized algorithms such as policy iteration, value iteration and their parallel versions to find e-op... |

127 | Probabilistic Roadmaps for Path Planning
- Kavraki, Svestka, et al.
- 1996
(Show Context)
Citation Context ...ecays to zero as the number of samples approaches infinity [76-83]. One of the first and most popular sampling-based algorithms is the Probabilistic RoadMap (PRM) algorithm proposed by Kavraki et al. =-=[50,77]-=-. The PRM algorithm first constructs an a priori graph, known as the roadmap, representing a rich set of collision-free trajectories and then answers multiple online queries by computing the shortest ... |

122 | Using randomization to break the curse of dimensionality
- Rust
- 1997
(Show Context)
Citation Context ...ces. For robotics applications where the state spaces are often unstructured due to cluttered environments or even unknown and dynamic, such global strategies are undesirable. Remarkably, as noted in =-=[27,48,49]-=-, algorithms based on random (or quasi-random) sampling of the state space, also known as sampling-based algorithms, provide a possibility to alleviate the curse of dimensionality when the control inp... |

117 |
Continuous-time stochastic control and optimization with financial applications
- Pham
- 2009
(Show Context)
Citation Context ...or engineering systems such as mechanical systems. Due to their abstraction, these models are also applied for problems in other domains such as mathematical economics [1, 2] and mathematical finance =-=[3]-=-. Therefore, stochastic optimal control has been studied extensively by several research communities, and each community focuses on different theoretical and implementation aspects of the field. Resea... |

107 | Probability methods for approximations in stochastic control and for elliptic equations - Kushner - 1977 |

98 |
Planning algorithms, Cambridge university press,
- LaValle
- 2006
(Show Context)
Citation Context ... planning problem is to find a sequence of feasible control inputs to drive the system from the initial state to one of the goal states and at the same time avoid collision with obstacles (see, e.g., =-=[52, 59, 60]-=- and references therein). The optimal version of the problem called optimal motion planning seeks for a feasible solution that minimizes some performance measure. These problems, which can be cast as ... |

95 | The Stochastic Motion Roadmap: A sampling framework for planning with Markov motion uncertainty
- Alterovitz, Simeon, et al.
- 2007
(Show Context)
Citation Context ...on different theoretical and implementation aspects of the field. Researchers also find applications of stochastic optimal control in diverse fields ranging from robotics [4], biology [5], healthcare =-=[6]-=- to management science, economics and finance [7,8]. In this thesis, we primarily focus on applications of stochastic optimal control in robotics, especially the problem of robot motion planning and c... |

87 | Probabilistic propositional planning: Representaions and complexity. - Littman - 1997 |

82 | Motion planning: a journey of robots, molecules, digital actors, and other - Latombe - 1999 |

78 | The Belief Roadmap: Efficient planning in belief space by factoring the covariance - Prentice, Roy - 2009 |

70 | Approaches for heuristically biasing RRT growth. - Urmson, Simmons - 2003 |

68 | LQR-trees: Feedback motion planning via sums-of-squares verification. - Tedrake, Manchester, et al. - 2010 |

60 |
An optimal one-way multigrid algorithm for discrete-time stochasticcontrol,”
- Chow, Tsitsiklis
- 1991
(Show Context)
Citation Context ...s and regularity theory for this class of PDEs are not well understood [38]. Thus, other methods approximate a continuous problem with a single discretetime finite-state Markov Decision Process (MDP) =-=[39,40]-=- without invoking the associated HJB equation. When dealing with finite-state MDPs, we can use specialized algorithms such as policy iteration, value iteration and their parallel versions to find e-op... |

60 | Dynamic motion planning for mobile robots using potential field method,”
- Ge, Cui
- 2002
(Show Context)
Citation Context ...an be applied to problems with state spaces of up-to five dimensions, cell decomposition methods suffer from the curse of dimensionality due to the large number of cells and difficult cell management =-=[68]-=-, and potential field methods suffer from local minima [69]. More importantly, all previously mentioned methods require an explicit representation of the obstacles in the configuration space for the c... |

58 | Incremental search methods for reachability analysis of continuous and hybrid systems - BHATIA, FRAZZOLI |

51 | Weak dynamic programming principle for viscosity solutions.
- Bouchard, Touzi
- 2011
(Show Context)
Citation Context ...programming principle encounters technical difficulties related to the measurable selection argument. Recently, in 2011, an approach called weak dynamic programming was proposed by Bouchard and Touzi =-=[21]-=- to derive HJB equations and find viscosity solutions that can avoid measurability issues. As shown in the authors' very recent works, the weak dynamic programming approach enables us to establish the... |

51 |
Numerical Methods for Stochastic Control
- Kushner, Dupuis
- 2001
(Show Context)
Citation Context ..., addresses several drawbacks of the previous methods. For the last three decades, Kushner, Dupuis, and their colleagues have developed a powerful method called Markov chain approximation (see, e.g., =-=[43, 44]-=- and references therein) to compute numerically value functions of a wide spectrum of stochastic optimal control problems. Conceptually, the Markov chain approximation method constructs a sequence of ... |

45 | Multiple RRTs for rapid replanning in dynamic environments - Zucker, Kuffner, et al. - 2007 |

44 | Real-Time Motion Planning with Applications to Autonomous Urban Driving.
- Kuwata, Teo, et al.
- 2009
(Show Context)
Citation Context ...cent years, several advanced autonomous systems have been built to operate in uncertain environments such as Mars rovers for planetary missions [9], autonomous cars providing urban mobility on demand =-=[10, 11]-=-, and small aerial vehicles operating in the presence of stochastic wind [12]. In many of these applications, the systems operate in worlds that are inherently continuous in time and space under a con... |

43 | Stochastic target problems, dynamic programming, and viscosity solutions, - Soner, Touzi - 2002 |

42 | Anytime RRTs. - Furgeson, Stentz - 2006 |

38 | Current issues in sampling-based motion planning - Lindemann, LaValle - 2003 |

38 | RRT-connect: An efficient approach to single-query path planning
- Jr, J, et al.
- 2000
(Show Context)
Citation Context ... algorithm called the Rapidly-Exploring Random Tree (RRT) were proposed by LaValle and Kuffner to avoid the need to specify a priori samples and tailored for single-query motion planning applications =-=[51,84,85]-=-. The RRT algorithm constructs a tree-based structure connecting an initial state to a goal region, which efficiently searches non-convex high dimensional search spaces. The algorithm is designed to d... |

38 | Sampling-based path planning on configuration-space costmaps - Jaillet, Cortés, et al. - 2010 |

37 | Connectivity of random k-nearest-neighbour graphs
- Ballister, Bollobás, et al.
- 2005
(Show Context)
Citation Context ...istance between the two points is bounded by a positive constant, we have Gilbert's disc model [101]. Another popular model called k-nearest neighbor graph considers edges between k nearest neighbors =-=[103]-=-. A remarkable result in this field certifies that when k = O(log n) where n is the number of points, the resulting graph is connected asymptotically almost surely and thus has optimal shortest paths ... |

31 | Manipulation planning among movable obstacles - Stilman, Schamburek, et al. - 2007 |

31 | Motion Planning of Aerial Robot using Rapidly-exploring Random Trees with Dynamic Constraints,” - Kim, Ostrowski - 2003 |

28 | RRTs for nonlinear, discrete, and hybrid planning and control,” in
- Branicky, Curtiss, et al.
- 2003
(Show Context)
Citation Context ... algorithm called the Rapidly-Exploring Random Tree (RRT) were proposed by LaValle and Kuffner to avoid the need to specify a priori samples and tailored for single-query motion planning applications =-=[51,84,85]-=-. The RRT algorithm constructs a tree-based structure connecting an initial state to a goal region, which efficiently searches non-convex high dimensional search spaces. The algorithm is designed to d... |

26 | Stochastic target problems with controlled loss. - Bouchard, Elie, et al. - 2009 |

25 | Real-time reach planning for animated characters using hardware acceleration - Liu, Badler |

22 | Limit theorems for stochastic processes, Theory Probab. - Skorokhod - 1956 |

22 | Bounding on rough terrain with the LittleDog robot - Shkolnik, Levashov, et al. - 2011 |

21 | Measure theoretic analysis of probabilistic path planning,” - Ladd, Kavraki - 2004 |

21 | Sampling-based planning, control and verification of hybrid systems - BRANICKY, CURTISS, et al. |

20 | An optimization approach to planning for mobile manipulation - Berenson, Kuffner, et al. |

17 |
Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE
- Touzi
- 2013
(Show Context)
Citation Context ...thod uses physical insights of the systems to construct the approximation. Interesting, given an elliptic PDE, it is possible to construct reversely an artificial stochastic dynamics for the equation =-=[24,44]-=-. Hence, Markov chain approximation is a probabilistic method to compute solutions of elliptic equations as well. Third, the method provides a mild sufficient condition, called local consistency, to c... |

17 | Computational approaches to drug design - Finn, Kavraki - 1999 |

16 | A voice-commandable robotic forklift working alongside humans in minimally-prepared outdoor environments,” in ICRA, - Teller, Walter, et al. - 2010 |

15 | Addressing cost-space chasms for manipulation planning,” in ICRA, - Berenson, Simeon, et al. - 2011 |

14 | K.L.: Numerical solution of Hamilton-Jacobi-Bellman equations by an upwind finite volume method - Wang, Jennings, et al. - 2003 |

14 | Randomized preprocessing of configuration for fast path planning - Kavraki, Latombe - 1994 |

13 | Optimal Control under Stochastic Target Constraints. - Bouchard, Elie, et al. - 2010 |

12 | Optimal continuous-parameter stochastic control - Fleming - 1969 |

12 | Weak dynamic programming for generalized state constraints, preprint
- Bouchard, Nutz
- 2011
(Show Context)
Citation Context ... ri) is equal to J*(z; rj) in OPT3. The constraint in the above formulation specifies the relationship of random variables at the terminal time as a target set, and hence the name of this formulation =-=[24,25]-=-2. In this formulation, we solve for feedback control policies o for all (z, r) E S instead of a particular choice of q for x(0) = z at time t = 0. We note that in this formulation, boundary condition... |

11 | Martingale methods in stochastic control, Stochastic Control Theory and Stochastic Differential Systems - Davis - 1979 |

11 | Bounds on tracking error using closed-loop rapidly-exploring random trees
- Luders, Karaman, et al.
- 2010
(Show Context)
Citation Context ... of constrained stochastic control problems are presented in Chapter 4. Bounded probabilities of failure In robotics, a common risk management problem is formulated as chance-constrained optimization =-=[75, 109--113]-=-. Historically, chance constraints specify that starting from a given initial state, the time- 0 probability of success must be above a given threshold where success means reaching goal areas safely. ... |

10 | Complexity of the mover’s problem and generalizations (extended abstract). - Reif - 1979 |

9 | Molecular disassembly with RRTlike algorithms,” in - Cortes, Jaillet, et al. - 2007 |

9 | Motion planning for highly constrained spaces - Yershova, LaValle - 2009 |

7 | Convergent numerical scheme for singular stochastic control with state constraints in a portfolio selection problem.
- Budhiraja, Ross
- 2007
(Show Context)
Citation Context ...rol, the HJB equations are in fact a system of partial differential inequalities. The existence, uniqueness of viscosity solutions and regularity theory for this class of PDEs are not well understood =-=[38]-=-. Thus, other methods approximate a continuous problem with a single discretetime finite-state Markov Decision Process (MDP) [39,40] without invoking the associated HJB equation. When dealing with fin... |

6 | Efficient collision checking in sampling-based motion planning. - Bialkowski, Karaman, et al. - 2012 |

4 | On heavy-tailed runtimes and restarts in rapidly-exploring random trees - Wedge, Branicky |

3 | A risk-constrained multi-stage decision making approach to the architectural analysis of planetary miissions
- Kuwata, Pavone, et al.
(Show Context)
Citation Context ... the problem of robot motion planning and control. In recent years, several advanced autonomous systems have been built to operate in uncertain environments such as Mars rovers for planetary missions =-=[9]-=-, autonomous cars providing urban mobility on demand [10, 11], and small aerial vehicles operating in the presence of stochastic wind [12]. In many of these applications, the systems operate in worlds... |

3 |
Dynamic Programming, 1st ed
- Bellman
- 1957
(Show Context)
Citation Context ...solutions to stochastic control problems. Since 1950, a variety of different approaches to stochastic optimal control have been investigated. Arguably, dynamic programming proposed by Bellman in 1957 =-=[13]-=- is one of the most well-known approaches. The dynamic programming principle provides a proper way to relate time-t optimal value function to any later time-T optimal value function. Bellman's princip... |

3 | Integration of path/maneuver planning in complex environments for agile maneuvering ucavs, - Koyuncu, Ure, et al. - 2009 |

3 | Free-configuration biased sampling for motion planning - Bialkowski, Otte, et al. - 2013 |

2 |
The markovdubins problem in the presence of a stochastic drift field
- Anderson, Bakolas, et al.
(Show Context)
Citation Context ...ertain environments such as Mars rovers for planetary missions [9], autonomous cars providing urban mobility on demand [10, 11], and small aerial vehicles operating in the presence of stochastic wind =-=[12]-=-. In many of these applications, the systems operate in worlds that are inherently continuous in time and space under a continuous control space. Moreover, we are often concerned with several aspects ... |

2 | Any-com collision checking: Sharing certificates in decentralized multi-robot teams - Otte, Bialkowski, et al. |

1 |
Stochastic optinial control and estimation methods adapted to the noise characteristics of the sensorimotor system
- Todorov
- 2005
(Show Context)
Citation Context ...mmunity focuses on different theoretical and implementation aspects of the field. Researchers also find applications of stochastic optimal control in diverse fields ranging from robotics [4], biology =-=[5]-=-, healthcare [6] to management science, economics and finance [7,8]. In this thesis, we primarily focus on applications of stochastic optimal control in robotics, especially the problem of robot motio... |

1 |
Stochastic optimial control, international finance and debt
- Fleming, Stein
- 2004
(Show Context)
Citation Context ...ts of the field. Researchers also find applications of stochastic optimal control in diverse fields ranging from robotics [4], biology [5], healthcare [6] to management science, economics and finance =-=[7,8]-=-. In this thesis, we primarily focus on applications of stochastic optimal control in robotics, especially the problem of robot motion planning and control. In recent years, several advanced autonomou... |

1 |
Robotic load balancing for mobility-on-denand systenis
- Pavone, Smith, et al.
- 2012
(Show Context)
Citation Context ...cent years, several advanced autonomous systems have been built to operate in uncertain environments such as Mars rovers for planetary missions [9], autonomous cars providing urban mobility on demand =-=[10, 11]-=-, and small aerial vehicles operating in the presence of stochastic wind [12]. In many of these applications, the systems operate in worlds that are inherently continuous in time and space under a con... |

1 |
Full bang to reduce predicted miss is optimal
- Benes
- 1976
(Show Context)
Citation Context ...ng conditions under which HJB equations have solutions (see survey in [14-18]). Establishing such conditions often limits the class of problems that can be handled by the dynamic programming approach =-=[19]-=-. In particular, these conditions allow value functions to be smooth enough so that they satisfy HJB equations in the classical or usual sense. However, in practice, value functions are often not smoo... |

1 | Encyclopaedia of complexity results for finite-horizon markov decision process problems - Mundhenk, Goldsmith, et al. - 1997 |

1 | An adaptive grid scheme for the discrete hamilton-jacobi-bellman equation - Grine - 1997 |

1 | The finite element approximation of hanilton-jacobi-bellman equations: the noncoercive case - Boulbrachene, Chentouf - 2004 |

1 | A random sampling scheme for path planning," in Robotics Research - Barraquand, Kavraki, et al. - 2000 |

1 |
Optimizations for sampling-based motion planning algorithms
- J
- 2013
(Show Context)
Citation Context ...rection is to combine parallelization and the interdependence of primitive procedures in the iMDP algorithm to speed up its running time. This direction is similar to the ideas proposed by Bialkowski =-=[97]-=- for RRT-like algorithms. 6.2.7 Collaborative multi-agent systems We can further consider a team of separate and independent agents collaborating to optimize a common objective function in uncertain e... |

1 |
Random graphs. Cambridge university press
- Bollobais
(Show Context)
Citation Context ...esses, and random geometric graphs. During our discussion in the next chapters, we will remind these notations, definitions, and results when necessary. The details of these materials can be found in =-=[43,104,128,129]-=-. 2.1 Basic Definitions and Notations Convergence We denote N as the set of natural numbers starting from 1, No = N U {0}, and R as the set of real numbers. Similarly, Rk is the set of k-dimensional r... |

1 |
Constrained miarkov decision models with weighted discounted rewards
- Feinberg, Feinberg, et al.
- 1993
(Show Context)
Citation Context ...ogy management and production management. The considered bounded trajectory performance constraints also have the same structure as the objective function with possibly different discount factors. In =-=[107,108]-=-, Feinberg and Shwartz consider these problems when constraints are applied for particular initial states. Thus, optimal control policies depend on the initial state. For this class of problems, the a... |

1 |
Constrained dynaic programning with two discount factors: applications and an algorithm," Automatic Control
- Feinberg, Shwartz
- 1999
(Show Context)
Citation Context ...ogy management and production management. The considered bounded trajectory performance constraints also have the same structure as the objective function with possibly different discount factors. In =-=[107,108]-=-, Feinberg and Shwartz consider these problems when constraints are applied for particular initial states. Thus, optimal control policies depend on the initial state. For this class of problems, the a... |