I would suggest to solve firstly the problem with a direct method, also called first-discretize-then-optimize methods. Objective function and ODEs as well as inequality constraints are discretized by simple methods, e.g. explicit or implicit Euler with fixed step size. Then solve the resulting NLP problem. Good software packages can be obtained from Matthias Gerdts (Munich), Christoph Bueskens (Bremen) or John Betts (Seattle). Or simply use the programming language AMPL in connection with IPOPT of Waechter and Biegler. The latter one is open source, while AMPL costs a little bit. We have made good experiences with the latter one.

In case all your controls enter objective function and constraints (ODEs) linearly you may not only have bang-bang, but also singular controls. The combination of AMPL and IPOPT provides you with estimates of the adjoint variables, which have to be scaled (!). By these together with the approximate state and control variables you can verify the neccessary conditions (e.g. the maximum, resp. minimum principle

a posteriori) yielding a usually extremely good guess for the switching structure. From a mathematical point of view you then have obtained an approximate candidate for a local optimal solution, not more.

If you are interested in even more precise results rewrite the optimal control problem as a switching point optimization problem, since you have a very good initial guess of the switching structure from the solution methods as described above; see.e.g., my papers.

If your problem is very complex, i.e. if you have many ODEs and many switching points I would not suggest to switch over to an indirect method such as multiple shooting unless you really need extremely precise results, for example for applications in aeronautics. (I was one of the first who have used multiple shooting

for complex optimal control problems in aeronautics.)

Finally, I never would recommend to use a genetic algorithm in such cases. Your proble has enough structure, which can be exploited by sophisticated methods.

Genetic algorithms would be too time consuming.

Good luck,

Hans Josef Pesch

We use dynamic programming in combination with PMP to solved the hybrid problem.

Example: Optimal Control of the Gearshift Command for Hybrid Electric Vehicles

Published in: Optimal Control of the Gearshift Command for Hybrid Electric Vehicles

Vehicular Technology, IEEE Transactions on (Volume:61 , Issue: 8 )

Date of Publication: Oct. 2012

Page(s): 3531 - 3543 ISSN : 0018-9545 INSPEC Accession Number: 13038894 Digital Object Identifier :10.1109/TVT.2012.2207922Date of Publication : 10 juli 2012 Date of Current Version : 12 oktober 2012 Issue Date : Oct. 2012 Sponsored by :IEEE Vehicular Technology Society

We used genetic algorithms for optimal controller equation’ search for a ship control. View a brief publication.

https://www.researchgate.net/publication/235763419_Evolutionary_algorithms_application_in_a_ship_autopilot_system_with_optimal_controller?ev=prf_pub

Conference Paper Evolutionary algorithms application in a ship autopilot syst...

We use Differential Evolution (which is in my my opinion more appropriate for continuous problems) to solve optimal control problems with several control variables and some degree of non-linearities in the system. It is computationaly very expensive but we are able to outperform (in terms of objective function value) the numerical techniques based on LQ framework. http://ideas.repec.org/a/eee/dyncon/v37y2013i4p821-837.html

I would suggest to solve firstly the problem with a direct method, also called first-discretize-then-optimize methods. Objective function and ODEs as well as inequality constraints are discretized by simple methods, e.g. explicit or implicit Euler with fixed step size. Then solve the resulting NLP problem. Good software packages can be obtained from Matthias Gerdts (Munich), Christoph Bueskens (Bremen) or John Betts (Seattle). Or simply use the programming language AMPL in connection with IPOPT of Waechter and Biegler. The latter one is open source, while AMPL costs a little bit. We have made good experiences with the latter one.

In case all your controls enter objective function and constraints (ODEs) linearly you may not only have bang-bang, but also singular controls. The combination of AMPL and IPOPT provides you with estimates of the adjoint variables, which have to be scaled (!). By these together with the approximate state and control variables you can verify the neccessary conditions (e.g. the maximum, resp. minimum principle

a posteriori) yielding a usually extremely good guess for the switching structure. From a mathematical point of view you then have obtained an approximate candidate for a local optimal solution, not more.

If you are interested in even more precise results rewrite the optimal control problem as a switching point optimization problem, since you have a very good initial guess of the switching structure from the solution methods as described above; see.e.g., my papers.

If your problem is very complex, i.e. if you have many ODEs and many switching points I would not suggest to switch over to an indirect method such as multiple shooting unless you really need extremely precise results, for example for applications in aeronautics. (I was one of the first who have used multiple shooting

for complex optimal control problems in aeronautics.)

Finally, I never would recommend to use a genetic algorithm in such cases. Your proble has enough structure, which can be exploited by sophisticated methods.

Genetic algorithms would be too time consuming.

Good luck,

Hans Josef Pesch

I used GA to optimize controller gains in AFSMC.

https://www.researchgate.net/publication/236659227_Genetic_Algorithms-based_Adaptive_Neural_Network_Fuzzy_Sliding_Mode_Control_for_MEMS_Angular_Rate_Sensor

Conference Paper Genetic Algorithms-based Adaptive Neural Network Fuzzy Slidi...

The use of Pontryagin’s maximum principle in optimal control theory can be found in:

Guan, J., Zhang, A. and Aral, M. M. (2013). “An Optimization Approach for Sustainable Development Planning of Savanna Systems,” Journal of Arid Environments, Vol. 98, pp. 60-69: DOI:10.1016/j.jaridenv.2013.07.013.

One does not need to use GA but that is also possible...

Hi,

If you can formulaze your system as an optimization problem, then, you can do it easily.

Regards,

Dr. Mohamed Arezki MELLAL

Thanks for sharing your ideas with me. They are helpful and will assist my team as we proceed.

I would only add that it has become the unfortunate habit to confuse "optimal" with "extremal" i.e. only satisfaction with necessary condition such as the Minimum Principle.. If one cannot prove optimality e.g. via sufficiency conditions or via existence and then comparison, one should at least state that one has only candidates for optimal solutions.

George Leitmann

GA can be applied to constrained optimisation problems. However, the success of it depends on how the chromosomes are coded and meaning fully decided. The Lagrangian function combining the objective and constraints can be evaluated by chromosome that would coded to represent the decision variables of the problem and Lagrangian Multipliers.

Prof. Leitmann - thank you for your clarification between "optimal" and "extremal". I am looking to combine the Euler-Lagrange necessary conditions with a GA. I will review the examples for Extremal Control in chapter 12 of your book "The Calculus of Variations and Optimal Control".

Dear Professor Cohen,

Thank you for your kind response. I hate to be a nitpicker, but I believe that

the distinction between optimal and extremal is important. "optimal" implies

more than "extremal", yet is often used when only "extremal" has been has

been proved. This led me to consider conditions sufficient for optimality,

as have others, of course.

Thank you again. Best regards, George Leitmann

Hello,

See the book Codex Biogenesis

www.amazon.fr/Codex-Biogenesis-13-codes-lADN/dp/2874340448/ref=sr_1_fkmr0_2?ie=UTF8&qid=1385628534&sr=8-2-fkmr0&keywords=codex+genesis%2C+perez

unfortunately not yet translated, but the author, my friend Jean Claude Perez, had different lectures in English he could provide you.

As ever in a thinking world,

Jacques De Schryver

Hi,

In complex systems is not easy to be applied the Pontryagin’s principle and to determine analytical optimal solution. In this case you could use the following strategy. First, for 20 -30 points of the domain control you can utilize the dynamical programming. For the time minimization the Bellman principle is true and you can utilize the dynamical programming and you can determine a first approximation of the control solution.

This solution is input of the second numerical algorithm- non-linear optimal control programming (for example Tabak and Kuo , Optimal control by mathematical programming, 1971, or something more new). The solution is not analytic but you can determine a numeric approximation of the optimal control.

Another possibility is to transform by the differential geometric technique the initial system to a equivalent simple system and to use the Pontryagin principle. For example if your system is observable you can transform the system to a Brunovsky normal form. For this form you can use the maximum principle step by step.

I totally agree Prof. G Leitmann, the classification is an important issue to consider, that will differentiate between optimal solution based on satisfying necessary and sufficient conditions and suboptimal solution based on using any type of approximations. Even for complex systems, the optimal setup for the optimal solution is possible; however the solution of that setup is another issue.

Dear Dr. Maher,

Thank you,

I would add that Prof. Pesch's suggestions are very useful. Indeed, the

check via sufficiency conditions is often easier, or even possible, for

the discretized system.

Sincerely, George Leitmann

A comment on the 'optimal' and 'extremal' solution. Sometimes it is 'easy' to prove the convexity of the problem, then you are sure that the candidate solution is a minimum. Otherwise you have to check if the second variation of your functional is positive. It is a huge computation but some books explain how to do it, e.g. prof. Hull's Optimal Control Theory for Applications, 2003.

Dear Dr. Frego,

You are quite right. Even use of the second variation pertains only to local and not global optimlilty. Indeed, analytical sufficiency conditions are often very restrictive and/ or hard to apply. 

My point is, unless optimality , local or global,  is proved, honesty requires to point this out and to claim only that the stated solution is a candidate for optimality.

This, in itself is important since it rules out solutions which do not satisfy necessary conditions such as the Maximum Principle.

In passing, since the Bellman 0ptimailty Principle has been invoked, it should be noted that it is a Theorem subject to conditions.

George Leitmann