| Fundamental concepts | p. 1 |
| Introduction | p. 1 |
| Fundamental definitions and notation | p. 2 |
| Operator | p. 2 |
| Vectors and matrices | p. 3 |
| Differentiation of a vector | p. 5 |
| Taylor series expansion | p. 5 |
| Norm of a vector | p. 6 |
| Sign definite | p. 6 |
| Stationary and maxima (minima) points | p. 7 |
| Steady-state system model | p. 7 |
| Continuous-time system model | p. 7 |
| Discrete-time system model | p. 8 |
| The performance index | p. 10 |
| Interpretation of results | p. 11 |
| Examples of systems for optimal control | p. 11 |
| Linear gas absorber | p. 11 |
| Nonlinear continuous stirred tank reactor | p. 13 |
| Photochemical reaction in CSTR | p. 15 |
| Production of secreted protein in a fed-batch reactor | p. 16 |
| Solving algebraic equations | p. 17 |
| Separation of the equations into two groups | p. 18 |
| Numerical examples | p. 19 |
| Application to multicomponent distillation | p. 31 |
| Solving ordinary differential equations | p. 32 |
| References | p. 32 |
| Steady-state optimization | p. 35 |
| Introduction | p. 35 |
| Linear programming | p. 35 |
| Example - diet problem with 5 foods | p. 38 |
| Interpretation of shadow prices | p. 42 |
| LJ optimization procedure | p. 44 |
| Determination of region size | p. 46 |
| Simple example - 5 food diet problem | p. 48 |
| Model reduction example | p. 48 |
| Parameter estimation | p. 54 |
| Handling equality constraints | p. 58 |
| References | p. 64 |
| Dynamic programming | p. 67 |
| Introduction | p. 67 |
| Examples | p. 67 |
| A simple optimal path problem | p. 68 |
| Job allocation problem | p. 69 |
| The stone problem | p. 72 |
| Simple optimal control problem | p. 73 |
| Linear optimal control problem | p. 75 |
| Cross-current extraction system | p. 76 |
| Limitations of dynamic programming | p. 80 |
| References | p. 80 |
| Iterative dynamic programming | p. 81 |
| Introduction | p. 81 |
| Construction of time stages | p. 82 |
| Construction of grid for x | p. 82 |
| Allowable values for control | p. 82 |
| First iteration | p. 82 |
| Stage P | p. 83 |
| Stage P - 1 | p. 83 |
| Continuation in backward direction | p. 83 |
| Iterations with systematic reduction in region size | p. 84 |
| Example | p. 85 |
| Use of accessible states as grid points | p. 85 |
| Algorithm for IDP | p. 86 |
| Early applications of IDP | p. 89 |
| References | p. 90 |
| Allowable values for control | p. 91 |
| Introduction | p. 91 |
| Comparison of uniform distribution to random choice | p. 92 |
| Uniform distribution | p. 93 |
| Random choice | p. 94 |
| References | p. 98 |
| Evaluation of parameters in IDP | p. 99 |
| Introduction | p. 99 |
| Number of grid points | p. 100 |
| Bifunctional catalyst blend optimization problem | p. 100 |
| Photochemical CSTR | p. 104 |
| Multi-pass approach | p. 106 |
| Nonlinear two-stage CSTR system | p. 107 |
| Further example | p. 109 |
| Effect of region restoration factor [eta] | p. 111 |
| Effect of the region contraction factor [gamma] | p. 112 |
| Effect of the number of time stages | p. 112 |
| References | p. 117 |
| Piecewise linear control | p. 119 |
| Introduction | p. 119 |
| Problem formulation | p. 119 |
| Algorithm for IDP for piecewise linear control | p. 120 |
| Numerical examples | p. 122 |
| Nonlinear CSTR | p. 122 |
| Nondifferentiable system | p. 124 |
| Linear system with quadratic performance index | p. 126 |
| Gas absorber with a large number of plates | p. 136 |
| References | p. 138 |
| Time-delay systems | p. 139 |
| Introduction | p. 139 |
| Problem formulation | p. 140 |
| Examples | p. 140 |
| Nonlinear two-stage CSTR system | p. 148 |
| References | p. 148 |
| Variable stage lengths | p. 149 |
| Introduction | p. 149 |
| Variable stage-lengths when final time is free | p. 155 |
| IDP algorithm | p. 156 |
| Problems where final time is not specified | p. 157 |
| Oil shale pyrolysis problem | p. 158 |
| Modified Denbigh reaction scheme | p. 161 |
| Systems with specified final time | p. 165 |
| Fed-batch reactor | p. 168 |
| References | p. 175 |
| Singular control problems | p. 177 |
| Introduction | p. 177 |
| Four simple-looking examples | p. 178 |
| Yeo's singular control problem | p. 191 |
| Nonlinear two-stage CSTR problem | p. 193 |
| References | p. 197 |
| State constraints | p. 199 |
| Introduction | p. 199 |
| Final state constraints | p. 199 |
| Problem formulation | p. 199 |
| Quadratic penalty function with shifting terms | p. 200 |
| Absolute value penalty function | p. 218 |
| Remarks on the choice of penalty functions | p. 223 |
| State inequality constraints | p. 224 |
| Problem formulation | p. 224 |
| State constraint variables | p. 225 |
| References | p. 234 |
| Time optimal control | p. 237 |
| Introduction | p. 237 |
| Time optimal control problem | p. 237 |
| Direct approach to time optimal control | p. 238 |
| Examples | p. 239 |
| Bridge crane system | p. 239 |
| Two-link robotic arm | p. 241 |
| Drug displacement problem | p. 243 |
| Two-stage CSTR system | p. 246 |
| High dimensional systems | p. 251 |
| References | p. 253 |
| Nonseparable problems | p. 255 |
| Introduction | p. 255 |
| Problem formulation | p. 255 |
| Examples | p. 256 |
| Luus-Tassone problem | p. 256 |
| Li-Haimes problem | p. 264 |
| References | p. 264 |
| Sensitivity considerations | p. 265 |
| Introduction | p. 265 |
| Example: Lee-Ramirez bioreactor | p. 267 |
| Solution by IDP | p. 268 |
| References | p. 272 |
| Toward practical optimal control | p. 273 |
| Introduction | p. 273 |
| Optimal control of oil shale pyrolysis | p. 274 |
| Future directions | p. 278 |
| References | p. 282 |
| Nonlinear algebraic equation solver | p. 283 |
| Program listing | p. 283 |
| Output of the program | p. 285 |
| Listing of linear programming program | p. 287 |
| Main program for the diet problem | p. 287 |
| Input subroutine | p. 288 |
| Subroutine for maximization | p. 289 |
| Output subroutine | p. 290 |
| LJ optimization programs | p. 291 |
| Five food diet problem | p. 291 |
| Model reduction problem | p. 293 |
| Geometric problem | p. 295 |
| Iterative dynamic programming programs | p. 297 |
| CSTR with piecewise constant control | p. 297 |
| IDP program for piecewise linear control | p. 303 |
| IDP program for variable stage lengths | p. 308 |
| Listing of DVERK | p. 315 |
| DVERK | p. 315 |
| Index | p. 321 |
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