In the real context any dynamic system can be represented by a set of differential equations of anty order. From these equations we can characterize the system and identify its intrinsic parameters. But what about the differential inclusions of first and second orde. Do anyone know any real application of the differential inclusions of first and second order?
See the following works :
- A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators. Paden B.E, Sastry S.S. Circuits and Systems, IEEE Transactions on (Volume:34 , Issue: 1 )
- https://www.researchgate.net/publication/261086915_Application_of_impulse_differential_inclusion_for_uncertainty_analysis_of_mechatronic_hybrid_system
Conference Paper Application of impulse differential inclusion for uncertaint...
See the following works :
- A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators. Paden B.E, Sastry S.S. Circuits and Systems, IEEE Transactions on (Volume:34 , Issue: 1 )
- https://www.researchgate.net/publication/261086915_Application_of_impulse_differential_inclusion_for_uncertainty_analysis_of_mechatronic_hybrid_system
Conference Paper Application of impulse differential inclusion for uncertaint...
http://www.zfm.ethz.ch/e/res/ove/index.htm
I know that in the modeling of friction and impact in dynamic systems, differential inclusions are widely used. Besides, differential inclusions are also used to model electrical circuits with diodes and other parts.
See the mainpage for a nice software Siconos developed for the simulation of such systems for more examples
http://siconos.gforge.inria.fr/
Dear researchers HAMOU and MAOYING;
I would like to thank you very much for your valuable references. The documents you refered to are very instructive.
many thanks;
Avoid differential inclusion at all cost. It can be dealt with game theory most of the time.
Thank you Pr. Zigang;
I know that the differential inclusion use requires a high performance computing models, At this time, I think that the parallel computing approach is well suited..
In differential inclusion, there is no existence and uniqueness results on the solution. You don't know what you are computing!
If the problem is viewed as an implicit system, the suitable solution looked for can be easily sorted out from the existing solutions;
Dear Professor Bouattane,
any dynamical system whose the external influence on the process is known up to some degree of uncertainty can be represented by a differential inclusion with the nonlinearity given by a multivalued map. For instance in R a multivalued map F can be viewed as an interval with extreme points two real functions, namely F(x)=[f(x),g(x)]. If the function f is lower semicontinuous and the function g is upper semicontinuous then the multivalued map F is upper semicontinuous in the multivalued sense, a usual regularity assumption needed to obtain existence results for the differential problem associated. Moreover if the dynamical system presents jumps it is possible to consider Fylippov type solutions. In the periodic treatment of some diseases, jumps may correspond to administration of a drug treatment; in environmental sciences, jumps may correspond to seasonal changes or harvesting; in economics jumps may correspond to abrupt changes of prices. As reference see Deimling, Multivalued differential equations and Hu, Papageorgiou, Handbook of Multivalued Analysis, Vol II.
Best regards,
Irene Benedetti
I agree with Zigang Pan. Most of the time (under sufficiently general conditions on paramaters of the problem), one has existence of solutions but uniqueness generally remains an unsolved problem. Besides existence one has to develop suitable criteria for selecting the correct "physical" solution.
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