Generalized Characteristics of First Order PDEs: Applications in Optimal Control and Differential GamesSpringer Science & Business Media, 15.05.1998 - 310 Seiten In some domains of mechanics, physics and control theory boundary value problems arise for nonlinear first order PDEs. A well-known classical result states a sufficiency condition for local existence and uniqueness of twice differentiable solution. This result is based on the method of characteristics (MC). Very often, and as a rule in control theory, the continuous nonsmooth (non-differentiable) functions have to be treated as a solutions to the PDE. At the points of smoothness such solutions satisfy the equation in classical sense. But if a function satisfies this condition only, with no requirements at the points of nonsmoothness, the PDE may have nonunique solutions. The uniqueness takes place if an appropriate matching principle for smooth solution branches defined in neighboring domains is applied or, in other words, the notion of generalized solution is considered. In each field an appropriate matching principle are used. In Optimal Control and Differential Games this principle is the optimality of the cost function. In physics and mechanics certain laws must be fulfilled for correct matching. A purely mathematical approach also can be used, when the generalized solution is introduced to obtain the existence and uniqueness of the solution, without being aimed to describe (to model) some particular physical phenomenon. Some formulations of the generalized solution may meet the modelling of a given phenomenon, the others may not. |
Inhalt
III | 7 |
IV | 8 |
V | 9 |
VI | 11 |
VIII | 13 |
IX | 17 |
X | 18 |
XI | 19 |
LXXX | 153 |
LXXXI | 155 |
LXXXII | 158 |
LXXXIII | 159 |
LXXXIV | 162 |
LXXXV | 164 |
LXXXVI | 168 |
LXXXVII | 169 |
XIII | 21 |
XIV | 24 |
XV | 27 |
XVI | 28 |
XVII | 29 |
XVIII | 31 |
XIX | 32 |
XX | 34 |
XXI | 35 |
XXII | 38 |
XXIV | 39 |
XXV | 41 |
XXVI | 52 |
XXVII | 54 |
XXVIII | 55 |
XXX | 57 |
XXXI | 60 |
XXXII | 62 |
XXXIII | 64 |
XXXVI | 66 |
XXXVII | 68 |
XXXVIII | 71 |
XXXIX | 72 |
XL | 74 |
XLI | 77 |
XLII | 80 |
XLIII | 84 |
XLIV | 85 |
XLV | 86 |
XLVI | 88 |
XLVII | 90 |
XLVIII | 91 |
XLIX | 93 |
L | 96 |
LI | 97 |
LII | 99 |
LIV | 102 |
LV | 103 |
LVI | 105 |
LVII | 106 |
LVIII | 109 |
LIX | 111 |
LX | 112 |
LXI | 115 |
LXII | 116 |
LXIII | 118 |
LXIV | 120 |
LXV | 123 |
LXVI | 126 |
LXVII | 127 |
LXVIII | 130 |
LXIX | 131 |
LXX | 132 |
LXXI | 135 |
LXXII | 137 |
LXXIII | 138 |
LXXIV | 139 |
LXXV | 140 |
LXXVI | 141 |
LXXVII | 142 |
LXXVIII | 145 |
LXXIX | 148 |
LXXXVIII | 173 |
LXXXIX | 175 |
XC | 176 |
XCI | 178 |
XCII | 182 |
XCIII | 184 |
XCV | 188 |
XCVI | 190 |
XCVII | 194 |
XCVIII | 195 |
XCIX | 198 |
C | 199 |
CI | 201 |
CIII | 203 |
CIV | 204 |
CV | 205 |
CVI | 208 |
CVII | 210 |
CVIII | 212 |
CIX | 213 |
CX | 214 |
CXI | 217 |
CXII | 220 |
CXIV | 222 |
CXV | 223 |
CXVI | 225 |
CXVII | 227 |
CXVIII | 228 |
CXIX | 231 |
CXXI | 234 |
CXXII | 236 |
CXXIII | 241 |
CXXIV | 243 |
CXXV | 245 |
CXXVI | 250 |
CXXVIII | 252 |
CXXIX | 253 |
CXXX | 255 |
CXXXI | 257 |
CXXXII | 258 |
CXXXIII | 261 |
CXXXIV | 263 |
CXXXV | 264 |
CXXXVI | 267 |
CXXXVII | 271 |
CXXXVIII | 272 |
CXXXIX | 276 |
CXL | 277 |
CXLI | 280 |
CXLII | 281 |
CXLIII | 282 |
CXLIV | 284 |
CXLV | 287 |
CXLVI | 289 |
CXLVII | 291 |
CXLVIII | 292 |
CXLIX | 293 |
CL | 296 |
CLI | 298 |
CLII | 301 |
308 | |
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Generalized Characteristics of First Order PDEs: Applications in Optimal ... Arik Melikyan Eingeschränkte Leseprobe - 2012 |
Generalized Characteristics of First Order PDEs: Applications in Optimal ... Arik Melikyan Keine Leseprobe verfügbar - 2012 |
Häufige Begriffe und Wortgruppen
A₁ B₁ Bellman equation boundary conditions Cauchy problem Chapter characteristic field codimension components considered construction coordinates corresponding curve D₁ defined denoted derivatives Differential Games dispersal surface equality equation equivocal surface Euler equation exists F₁ F₂ FF₁}F Figure formulation fulfilled function u(x game value geodesic line gradient Hamilton-Jacobi equation Hamiltonian hypersurface inequality initial conditions integral surface Jacobi bracket Lemma linear manifold necessary conditions nonzero notations obtain ODE system Optimal Control optimal paths order PDE p₁ parameters player primary solution regular characteristics regular paths relations satisfies scalar secondary domain segment self-similar shock wave singular characteristics singular paths singular surface smooth smooth function system of singular T₁ tangent Theorem trajectories twice differentiable u₁ unique uo(x value problem vanish variation vector viscosity solution Y₁ მე მთ მყ
Beliebte Passagen
Seite 2 - Traditionally, only the field on the surface of codimension one was associated with first order PDEs, the nonsmoothness of the solution and/or of the Hamiltonian being considered as an obstacle for the application of the classical MC. The main finding of this book says that one can overcome this obstacle using the same characteristics, but defined on the appropriate surface of codimension...
Seite 2 - They were found due to investigation of singular paths in Differential Games and Optimal Control. Regular paths in these domains are known to be governed by the Hamiltonian ODE-system, the characteristic system for HJBI-equation. Singular paths for many cases are described by similar equations using the so-called singular controls.
Seite 2 - This field increases by one the dimension of the initial strip preserving its strip property. Such a field can be defined for the even-dimensional surfaces of an odd codimension 1,3, There exists the differential-geometric definition of the characteristic field, embracing all this cases. Traditionally...
Seite 2 - The differential-geometric description of regular and singular characteristics is the following. The classical characteristics define a tangent field on the even-dimensional hypersurface (of codimension one) corresponding to the PDE. More precisely, it is a 2n-dimensional surface in the (In + 1)dimensional space of the state and co-state vectors plus one scalar variable (the function itself).
Seite 2 - The integral curves of the characteristic fields on the surfaces of codimension 3 and more are called, singular characteristics. Thus, two types of characteristics, regular and singular, are associated with a nonlinear PDE.
Seite 2 - ... so-called singular controls. The attempt to eliminate singular controls from these equations has led to the discovery of singular characteristics, which appear to be inherent not only to game or control problems but rather to general nonlinear first order PDE.
Seite 2 - Thus, two types of characteristics, regular and singular, are associated with a nonlinear PDE. In many cases using both regular and singular characteristics one can construct the solution if the latter itself and/or the Hamiltonian is nonsmooth.
Verweise auf dieses Buch
Handbook of First-Order Partial Differential Equations Andrei D. Polyanin,Valentin F. Zaitsev,Alain Moussiaux Eingeschränkte Leseprobe - 2001 |
Max-Plus Methods for Nonlinear Control and Estimation William M. McEneaney Eingeschränkte Leseprobe - 2006 |