Difference between revisions of "Category:ODE model"
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| − | This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only (typically time) are present in the | + | This category includes all problems constrained by the solution of [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equations] (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only (typically time) are present in the [[:Category:Model characterization|model equations]] for <math>F(\cdot)</math>. |
The mixed-integer optimal control problem is of the form | The mixed-integer optimal control problem is of the form | ||
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== Extensions == | == Extensions == | ||
| − | For some problems the functions may as well depend explicitely on the time <math>t</math>. | + | * For some problems the functions may as well depend explicitely on the time <math>t</math>. |
| − | + | * The differential equations might depend on [[:Category:State dependent switches | state-dependent switches]]. | |
| − | The differential equations might depend on [[:Category:State dependent switches | state-dependent switches]]. | + | * The variables may include [[:Category:Boolean variables | boolean variables]]. |
| + | * The underlying process might be a [[:Category:Multistage process | multistage process]]. | ||
| + | * The dynamics might be [[:Category:Unstable | unstable]]. | ||
| + | * There might be an underlying [[:Category:Network topology | network topology]]. | ||
Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional. | Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional. | ||
[[Category:Model characterization]] | [[Category:Model characterization]] | ||
Revision as of 17:38, 18 November 2010
This category includes all problems constrained by the solution of ordinary differential equations (ODE). In particular, no algebraic variables and derivatives with respect to one independent variable only (typically time) are present in the model equations for 
.
The mixed-integer optimal control problem is of the form
![\begin{array}{llcl}
\displaystyle \min_{x(\cdot), u(\cdot), v(\cdot), q, \rho} & \phi(x(t_f), q, \rho) \\[1.5ex]
\mbox{s.t.} & \dot{x}(t) & = & f(x(t), u(t), v(t), q, \rho), \\
& 0 &\le& c(x(t),u(t),v(t), q, \rho), \\[1.5ex]
& 0 &=& r^{\text{eq}}(x(t_0),x(t_1), \dots, x(t_m), q, \rho), \\
& 0 &\le& r^{\text{ieq}}(x(t_0),x(t_1), \dots, x(t_m), q, \rho), \\[1.5ex]
& v(t) &\in& \Omega := \{v^1, v^2, \dots, v^{n_\omega} \},\\
& \rho &\in& \Rho := \{\rho^1, \rho^2, \dots, \rho^{n_\Rho} \},
\end{array}](https://mintoc.de/images/math/c/2/e/c2ea6ba320c6ca71260f7eeb98105969.png)
for ![t \in [t_0, t_f]](https://mintoc.de/images/math/5/5/8/55823791d9100bcb5461801aff4f6edd.png)
almost everywhere.
denotes the differential states,
denotes the continuous control functions,
denotes the integer control functions,
denotes the continuous (constant-in-time) control values,
denotes the integer (constant-in-time) control values.
The multipoint constraints 
are defined on a time grid 
. The Mayer term functional 
, the path- and control constraints 
and the constraint functions 
are assumed to be sufficiently often differentiable.
The equality constraints 
will often fix the initial values, i.e., 
, or impose a periodicity constraint.
Extensions
- For some problems the functions may as well depend explicitely on the time
. - The differential equations might depend on state-dependent switches.
- The variables may include boolean variables.
- The underlying process might be a multistage process.
- The dynamics might be unstable.
- There might be an underlying network topology.
Note that a Lagrange term 
can be transformed into a Mayer-type objective functional.
Pages in category "ODE model"
The following 52 pages are in this category, out of 52 total.
