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15
Nonlinear optimal control via occupation measures and LMI relaxations
 SIAM Journal on Control and Optimization
, 2008
"... Abstract. We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control constraints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state an ..."
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Cited by 47 (24 self)
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Abstract. We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control constraints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI (linear matrix inequality)relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments. 1.
Convergence of a nonmonotone scheme for HamiltonJacobiBellman equations with discontinuous data
, 2007
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An efficient data structure and accurate scheme to solve front propagation problems
 J. Sci. Comput
"... In this paper, we are interested in some front propagation problems coming from control problems in ddimensional spaces, with d ≥ 2. As opposed to the usual level set method, we localize the front as a discontinuity of a characteristic function. The evolution of the front is computed by solving an ..."
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Cited by 10 (8 self)
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In this paper, we are interested in some front propagation problems coming from control problems in ddimensional spaces, with d ≥ 2. As opposed to the usual level set method, we localize the front as a discontinuity of a characteristic function. The evolution of the front is computed by solving an HamiltonJacobiBellman equation with discontinuous data, discretized by means of the antidissipative Ultra Bee scheme. We develop an efficient dynamic storage technique suitable for handling front evolutions in large dimension. Then we propose a fast algorithm, showing its relevance on several challenging tests in dimension d = 2, 3, 4. We also compare our method with the techniques usually used in level set methods. Our approach leads to a computational cost as well as a memory allocation scaling as O(Nnb) in most situations, where Nnb is the number of grid nodes around the front. AMS Classification: 65M06, 49L99.
A DISCONTINUOUS GALERKIN SCHEME FOR FRONT PROPAGATION WITH OBSTACLES
"... Abstract. We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski, Cheng and Shu [6]), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of Bokanowski et ..."
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Cited by 7 (2 self)
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Abstract. We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski, Cheng and Shu [6]), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of Bokanowski et al. [8], leading to a level set formulation driven by min(ut+H(x,∇u),u−g(x)) = 0, where g(x) is an obstacle function. The DG scheme is motivated by the variational formulation when the Hamiltonian H is a linear function of∇u, correspondingtolinear convectionproblems in presence ofobstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis are performed for the linear case with Euler forward, a Heun scheme and a RungeKutta third order time discretization using the technique proposed in Zhang and Shu [22]. Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost. 1.
A discontinuous Galerkin solver for front propagation
, 2009
"... Abstract. We propose a new discontinuous Galerkin (DG) method based on [9] to solve a class of HamiltonJacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical expe ..."
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Cited by 5 (3 self)
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Abstract. We propose a new discontinuous Galerkin (DG) method based on [9] to solve a class of HamiltonJacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical experiments show the relevance of our method, in particular for front propagation. The HamiltonJacobi (HJ) equation 1.
Capture Basin Approximation using Interval Analysis
 INT. J. ADAPT. CONTROL SIGNAL PROCESS. 2002; 00:1–6
, 2002
"... This paper proposes a new approach for computing the capture basin C of a target T. The capture basin corresponds to the set of initial state vectors such that the target could be reached in finite time via an appropriate control input, before possibly leaving the target. Whereas classical capture b ..."
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Cited by 4 (2 self)
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This paper proposes a new approach for computing the capture basin C of a target T. The capture basin corresponds to the set of initial state vectors such that the target could be reached in finite time via an appropriate control input, before possibly leaving the target. Whereas classical capture basin characterization do not provide any guarantee on the set of state vectors that belong to the capture basin, interval analysis and guaranteed numerical integration allow us to avoid any indetermination. We present an algorithm able to provide guaranteed approximation of the inner C − and an the outer C + of the capture basin, such that C − ⊆ C ⊂ C +. In order to illustrate the principle and the efficiency of the approach, a testcase on the ”car on the hill” problem is provided.
OPTIMAL VIABLE PATH SEARCH FOR A CHEESE RIPENING PROCESS USING A MULTIOBJECTIVE EA
"... Multiobjective Evolutionary aglorithm, viability modeling, optimal path search, indirect encoding, agrifood process modeling, cheese ripening. ..."
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Cited by 2 (0 self)
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Multiobjective Evolutionary aglorithm, viability modeling, optimal path search, indirect encoding, agrifood process modeling, cheese ripening.
INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING Int. J. Adapt. Control Signal Process. 2002; 00:1–6 Prepared using acsauth.cls [Version: 2002/11/11 v1.00] Capture Basin Approximation using Interval Analysis
"... This paper proposes a new approach for computing the capture basin C of a target T. The capture basin corresponds to the set of initial state vectors such that the target could be reached in finite time via an appropriate control input, before possibly leaving the target. Whereas classical capture b ..."
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This paper proposes a new approach for computing the capture basin C of a target T. The capture basin corresponds to the set of initial state vectors such that the target could be reached in finite time via an appropriate control input, before possibly leaving the target. Whereas classical capture basin characterization do not provide any guarantee on the set of state vectors that belong to the capture basin, interval analysis and guaranteed numerical integration allow us to avoid any indetermination. We present an algorithm able to provide guaranteed approximation of the inner C − and an the outer C + of the capture basin, such that C − ⊆ C ⊂ C+. In order to illustrate the principle and the efficiency of the approach, a testcase on the ”car on the hill ” problem is provided. Copyright c © 2002
An efficient data structure to solve front propagation problems
"... In this paper we develop a general efficient sparse storage technique suitable to coding front evolutions in d ≥ 2 space dimensions. This technique is mainly applied here to deal with deterministic target problems with constraints, and solve the associated minimal time problems. To this end we consi ..."
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In this paper we develop a general efficient sparse storage technique suitable to coding front evolutions in d ≥ 2 space dimensions. This technique is mainly applied here to deal with deterministic target problems with constraints, and solve the associated minimal time problems. To this end we consider an HamiltonJacobiBellman equation and use an adapted antidiffusive UltraBee scheme. We obtain a general method which is faster than a full storage technique. We show that we can compute problems that are out of reach by full storage techniques (because of memory). Numerical experiments are provided in dimension d = 2, 3, 4. Moreover, the application of the sparse storage technique to the implementation of the Fast Marching Method for the eikonal equation, in dimensions 2 and 3, is discussed. AMS Classification: 65M06, 49L99.