**6. Concluding remarks**

*<sup>K</sup>*ð Þ *best* <sup>¼</sup> *<sup>K</sup>*ð Þ *best*

*<sup>σ</sup> <sup>A</sup>* � *BK*ð Þ *best <sup>C</sup>* � � <sup>¼</sup>

*<sup>K</sup>*ð Þ *best* <sup>¼</sup> *<sup>K</sup>*ð Þ *best*

*<sup>σ</sup> <sup>A</sup>* � *BK*ð Þ *best <sup>C</sup>* � � <sup>¼</sup>

max <sup>¼</sup> <sup>1</sup>*:*<sup>1817</sup> � <sup>10</sup>6, *<sup>K</sup>*ð Þ *best* �

max <sup>¼</sup> <sup>5</sup>*:*<sup>2247</sup> � <sup>10</sup>4, *<sup>K</sup>*ð Þ *best* �

 *<sup>K</sup>*ð Þ *best* h i

*K*ð Þ *best* ¼

*Control Theory in Engineering*

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

*K*ð Þ *best* ¼

*σ*ð Þ *best*

*K*ð Þ *best* ¼

*K*ð Þ *best* ¼

*σ*ð Þ *best*

and by RS + MC Algorithms 1 and 2:

by the MC Algorithm 2

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

 *<sup>K</sup>*ð Þ *best* h i

>>>>>>>>><

>>>>>>>>>:

>>>>>>>>><

>>>>>>>>>:

� �

� �

, where

, where

*:*60115550 0*:*71670943 1*:*38518242

,

,

>>>>>>>>>=

,

,

,

>>>>>>>>>=

,

>>>>>>>>>;

>>>>>>>>>;

�1*:*21472623 7*:*41955425 �2*:*28748737

*:*08032866 �1*:*01678491 �3*:*50944968

�0*:*43606038 �0*:*52797919 �1*:*91992615 �0*:*37181168 �0*:*49434738 �1*:*97765275

�0*:*43167652 2*:*40298411 �0*:*14274690 �1*:*34628983 �2*:*73288946 �0*:*19682963

*:*86726666, 0*:*98939915,

*:*99600356 � 0*:*01886288*i*, *:*99670041 � 0*:*00199237*i*,

*:*97791267 � 0*:*01292326*i*,

� ¼ 8*:*1964,

*:*16219293 0*:*22685502 �1*:*41842788 *:*10558055 0*:*04299252 �1*:*12924145 �0*:*01638169 �0*:*08184317 �0*:*14133958

�0*:*03859594 0*:*07947463 0*:*12534983

�0*:*04123859 0*:*04014410 0*:*05631322 �0*:*06158264 0*:*05048672 0*:*16208075

> *:*53091472, 0*:*95364142, *:*98741833, 0*:*99897830,

*:*99717581 � 0*:*00564301*i*, *:*95033459 � 0*:*08956374*i*,

� ¼ 195*:*3621,

�0*:*23992446 �0*:*17536269 �0*:*44875450 �0*:*14481130 �0*:*18150136 �0*:*27764220

*:*88639191 �0*:*92895747 0*:*77107501

The Ray-Shooting Method is a powerful tool, since it practically solves the problem of LQR via SOF, for real-life discrete-time LTI systems. The proposed hybrid algorithm RS + MC has good performance in terms of run-time, in terms of the quality of controllers (by reducing the starting point LQR functional value and by reducing the controller norm) and in terms of the success rate in finding a starting point feasible with respect to the needed *α*-stability. The RS + MC algorithm has a proof of convergence in probability to a global minimum (as is evidently seen from the experiments). This enlarges the practicality and scope of the Ray-Shooting Method in solving hard complexity control problems, and we expect to receive more results in this direction.

*Control Theory in Engineering*
