CΔ,

Δ, where

h bð Þ <sup>1</sup> þ c<sup>1</sup> Δ

<sup>1</sup> a0H<sup>1</sup> is pro-

which gives <sup>a</sup><sup>1</sup> <sup>þ</sup> <sup>b</sup><sup>1</sup>′<sup>Δ</sup> <sup>þ</sup> ð Þ <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>1</sup> # <sup>a</sup><sup>1</sup> <sup>þ</sup> ð Þ ð Þ <sup>1</sup> <sup>þ</sup> Mh <sup>b</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>1</sup> # <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>b</sup><sup>0</sup> <sup>þ</sup> <sup>c</sup>0.

<sup>a</sup><sup>1</sup>″# M bð Þ <sup>1</sup>′ <sup>þ</sup> <sup>c</sup><sup>1</sup>′ <sup>Δ</sup>, <sup>b</sup><sup>2</sup> # M bð Þ <sup>1</sup>′ <sup>þ</sup> <sup>c</sup><sup>1</sup>′ <sup>Δ</sup> and <sup>c</sup><sup>2</sup> # M bð Þ <sup>1</sup>′ <sup>þ</sup> <sup>c</sup><sup>1</sup>′ <sup>Δ</sup>, where <sup>M</sup> as before con-

h bð Þ <sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>1</sup> <sup>Δ</sup> <sup>b</sup><sup>2</sup> # <sup>4</sup>MM<sup>~</sup> <sup>2</sup>

j j ak � ak�<sup>1</sup> <sup>0</sup> # <sup>x</sup><sup>k</sup>

Note, ð Þ ak; bk;ck is a Cauchy sequences which converges to ð Þ a∞; 0; 0 , where

∞ k¼0

exp ðð Þ <sup>K</sup><sup>1</sup>′AdKð Þ Xd <sup>K</sup><sup>1</sup> <sup>Δ</sup>Þ ¼ exp ð Þ �k<sup>1</sup>″ exp <sup>a</sup>0<sup>Δ</sup> <sup>þ</sup> <sup>C</sup>Δ<sup>2</sup> exp ð Þ �Ak<sup>2</sup>″A′ :

exp ð Þ AdKð Þ Xd <sup>Δ</sup> <sup>K</sup>1AK<sup>2</sup> <sup>¼</sup> KaH<sup>1</sup> exp <sup>a</sup><sup>0</sup>′<sup>Δ</sup> <sup>þ</sup> <sup>C</sup>′Δ<sup>2</sup> AH2Kb

<sup>1</sup> <sup>a</sup>0<sup>Δ</sup> <sup>þ</sup> <sup>C</sup>Δ<sup>2</sup> <sup>H</sup><sup>1</sup> <sup>¼</sup> <sup>a</sup><sup>0</sup>′<sup>Δ</sup> <sup>þ</sup> <sup>C</sup>′Δ<sup>2</sup> is in <sup>a</sup> and <sup>a</sup><sup>0</sup>′ <sup>¼</sup> P H�<sup>1</sup>

k

αkWkð Þ Xd :

where <sup>a</sup>0<sup>Δ</sup> <sup>þ</sup> <sup>C</sup>Δ<sup>2</sup> <sup>∈</sup> <sup>f</sup>. By using a stabilizer <sup>H</sup>1, H2, we can rotate them to <sup>a</sup> such

The above iterative procedure generates <sup>k</sup><sup>1</sup>′ and <sup>k</sup><sup>2</sup>″ in Eq. (47), such that

x<sup>k</sup> #

<sup>¼</sup> exp ð Þ �ð Þ <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>1</sup> <sup>Δ</sup> exp <sup>a</sup>1<sup>Δ</sup> <sup>þ</sup> ð Þ <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup><sup>1</sup> <sup>Δ</sup> <sup>þ</sup> <sup>b</sup>1′Δ<sup>2</sup>

<sup>¼</sup> exp <sup>a</sup>1<sup>Δ</sup> <sup>þ</sup> <sup>b</sup>1′Δ<sup>2</sup> <sup>þ</sup> <sup>c</sup>1′Δ<sup>2</sup>

¼ exp ð Þ a2Δ þ b2Δ þ c2Δ

where using the bound 2M Mh <sup>~</sup> ð Þ <sup>þ</sup> <sup>1</sup> <sup>Δ</sup><1, we obtain

where c<sup>1</sup> ∈ k, such that c<sup>1</sup> # Mhb1.

We can decompose, ð Þ <sup>b</sup><sup>1</sup>′ <sup>þ</sup> <sup>c</sup><sup>1</sup>′ <sup>Δ</sup><sup>2</sup>

hΔ< <sup>2</sup> 3

Using ð Þ bk <sup>þ</sup> ck # x bð Þ <sup>k</sup>�<sup>1</sup> <sup>þ</sup> ck�<sup>1</sup> # <sup>x</sup><sup>k</sup>ð Þ <sup>b</sup><sup>0</sup> <sup>þ</sup> <sup>c</sup><sup>0</sup> .

j j a<sup>∞</sup> � a<sup>0</sup> <sup>0</sup> # x bð Þ <sup>0</sup> þ c<sup>0</sup> ∑

P H�<sup>1</sup> <sup>1</sup> a0H<sup>1</sup> <sup>¼</sup> <sup>∑</sup>

h bð Þ <sup>0</sup> þ c<sup>0</sup> .

verts between two different norms.

This gives

Similarly,

<sup>a</sup><sup>2</sup> # <sup>a</sup><sup>1</sup> <sup>þ</sup> <sup>4</sup>MM<sup>~</sup> <sup>2</sup>

where <sup>C</sup> <sup>¼</sup> <sup>16</sup>MM<sup>~</sup> <sup>2</sup>

such that H�<sup>1</sup>

jection onto a such that

that

182

For <sup>x</sup> <sup>¼</sup> <sup>8</sup>MM<sup>~</sup> <sup>2</sup>

where Δ is chosen small.

Applied Modern Control

$$\exp\left(b\right) = K\_a \exp\left(a + a\_1 \Delta + \mathcal{W}(b\_1)\Delta\right) K\_{b\nu}$$

where Wð Þ b<sup>1</sup> is Weyl element of b1. Furthermore

$$\exp\left(b + b\_2\Delta\right) = K\_{a'} \exp\left(a + a\_1\Delta + \mathcal{W}(b\_1)\Delta + \mathcal{W}(b\_2)\Delta\right) K\_{b'} \dots$$

Proof. Note, <sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>K</sup>�<sup>1</sup> <sup>3</sup> PK�<sup>1</sup> <sup>4</sup> , commutes with b1. This implies

<sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>K</sup><sup>~</sup> exp ð Þ <sup>a</sup> <sup>þ</sup> <sup>a</sup>1<sup>Δ</sup> <sup>K</sup> commutes with <sup>b</sup>1. This implies <sup>A</sup>2b1A�<sup>1</sup> <sup>2</sup> ¼ b1, i.e., <sup>K</sup><sup>~</sup> exp ð Þ <sup>a</sup> <sup>þ</sup> <sup>a</sup>1<sup>Δ</sup> AdKð Þ <sup>b</sup><sup>1</sup> exp ð Þ �ð Þ <sup>a</sup> <sup>þ</sup> <sup>a</sup>1<sup>Δ</sup> <sup>K</sup><sup>~</sup> <sup>0</sup> <sup>¼</sup> <sup>b</sup>1, which implies that AdKð Þ <sup>b</sup><sup>1</sup> <sup>∈</sup>f. Recall, from Remark 5,

$$\exp\left(a + a\_1 \Delta\right) \mathrm{Ad}\_K(b\_1) \exp\left(-(a + a\_1 \Delta)\right) = \sum\_k c\_k (Y\_k \cos\left(\lambda\_k\right) + X\_k \sin\left(\lambda\_k\right)).$$

This implies ∑kck sin ð Þ λ<sup>k</sup> Xk ¼ 0, implying λ<sup>k</sup> ¼ nπ. Therefore,

$$\exp\left(2(a+a\_1\Delta)\right)Ad\_K(b\_1)\exp\left(-2(a+a\_1\Delta)\right) = Ad\_K(b\_1).$$

We have shown existence of <sup>H</sup><sup>1</sup> such that <sup>H</sup>1AdKð Þ <sup>b</sup><sup>1</sup> <sup>H</sup>�<sup>1</sup> <sup>1</sup> ∈a, using H1, H<sup>2</sup> as before,

$$\begin{aligned} \tilde{K} \exp\left(a + a\_1 \Delta\right) K \exp\left(b\_1 \Delta\right) &= \tilde{K} H\_2 \exp\left(a + a\_1 \Delta\right) H\_1 \exp\left(A d\_K(b\_1) \Delta\right) K \\ &= K\_d \exp\left(a + a\_1 \Delta + \mathcal{W}(b\_1) \Delta\right) K\_b .\end{aligned}$$

Applying the theorem again to

$$K\_a \exp\left(a + a\_1 \Delta + \mathcal{W}(b\_1)\Delta\right) K\_b \exp\left(b\_2 \Delta\right) = K\_{a^\*} \exp\left(a + a\_1 \Delta + \mathcal{W}(b\_1)\Delta + \mathcal{W}(b\_2)\Delta\right) K\_{b^\*} \cdot \mathcal{W}$$

Lemma 2 Given Pi <sup>¼</sup> <sup>K</sup><sup>i</sup> 1Ai Ki <sup>2</sup> <sup>¼</sup> <sup>K</sup><sup>i</sup> <sup>1</sup> exp ai � �K<sup>i</sup> 2, we have Pi,iþ<sup>1</sup> ¼ exp H<sup>þ</sup> <sup>i</sup> Δ<sup>þ</sup> i � �Pi, and Pi,iþ<sup>1</sup> ¼ exp �H� <sup>i</sup>þ<sup>1</sup>Δ� iþ1 � �Piþ1, where <sup>H</sup><sup>þ</sup> <sup>i</sup> ¼ AdKi ð Þ Xd . From above we can express

$$P\_{i,i+1} = K\_a^{i+} \exp\left(a^i + a\_1^{i+} \Delta\_+^i + a\_2^{i+} \left(\Delta\_+^i\right)^2\right) K\_b^{i+}.$$

where ai<sup>þ</sup> <sup>1</sup> and ai<sup>þ</sup> <sup>2</sup> are first and second order increments to ai in the positive direction. The remaining notation is self-explanatory.

$$P\_{i,i+1} = K\_a^{(i+1)-} \exp\left(a^{i+1} - a\_1^{(i+1)-} \Delta\_-^{i+1} - a\_2^{(i+1)-} \left(\Delta\_-^{i+1}\right)^2\right) K\_b^{(i+1)-}.$$

$$\exp\left(a^{i+1}\right) = K\_1 \exp\left(a^i + a\_1^{i+} \Delta\_+^i + a\_2^{i+} \left(\Delta\_+^i\right)^2 + \mathcal{W}\left(a\_1^{(i+1)-} \Delta\_-^{i+1} + a\_2^{(i+1)-} \left(\Delta\_-^{i+1}\right)^2\right)\right) K\_2.$$

$$\mathcal{W}\left(a\_1^{(i+1)-} \Delta\_-^{i+1} + a\_2^{(i+1)-} \left(\Delta\_-^{i+1}\right)^2\right) = \mathcal{P}\left(\mathcal{W}\left(a\_1^{(i+1)-}\right)\right) \Delta\_-^{i+1} + \mathcal{P}\left(\mathcal{W}\left(a\_2^{(i+1)-}\right)\right) \left(\Delta\_-^{i+1}\right)^2$$

$$= \sum\_k a\_k \mathcal{W}\_k(\mathcal{X}\_d) \Delta\_-^{i+1} + o\left(\left(\Delta\_-^{i+1}\right)^2\right)$$

where, a<sup>i</sup> , a<sup>i</sup> 1, a<sup>i</sup> <sup>2</sup> ∈a. Using Lemma 1 and 2, we can express

$$P\_n(T) = K\_1 \exp\left(a\_n\right) \exp K\_2 = K\_1 \exp\left(\sum\_i \mathcal{W}(a\_i^+) \Delta\_i^+ + \mathcal{W}(a\_{i+1}^-) \Delta\_{i+1}^-\right) \exp\left(\underbrace{\sum\_i \rho(\Delta^2)}\_{\leq nT}\right) K\_2$$

Letting ε go to 0, we have

$$P\_n(T) = K\_1 \exp\left(T \sum\_i \alpha\_i \mathcal{W}\_i(X\_d)\right) K\_2.$$

Hence the proof of theorem. q.e.d.

#### 4. Conclusion

In this chapter, we studied some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. We saw how dynamics was decomposed into fast generators k (local Hamiltonians) and slow generators p (couplings) as a Cartan decomposition g ¼ p ⊕ k. Using this decomposition, we used some convexity ideas to completely characterize the reachable set and time optimal control for these problems.

Author details

Navin Khaneja

185

Systems and Control Engineering, IIT Bombay, India

provided the original work is properly cited.

\*Address all correspondence to: navinkhaneja@gmail.com

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

DOI: http://dx.doi.org/10.5772/intechopen.80567

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics DOI: http://dx.doi.org/10.5772/intechopen.80567

### Author details

where ai<sup>þ</sup>

Applied Modern Control

W a

ð Þ� iþ1 <sup>1</sup> <sup>Δ</sup>iþ<sup>1</sup> � <sup>þ</sup> <sup>a</sup>

where, a<sup>i</sup>

4. Conclusion

problems.

184

<sup>1</sup> and ai<sup>þ</sup>

exp aiþ<sup>1</sup> � � <sup>¼</sup> <sup>K</sup><sup>1</sup> exp ai <sup>þ</sup> ai<sup>þ</sup>

, a<sup>i</sup> 1, a<sup>i</sup> <sup>2</sup> ∈a. Using Lemma 1 and 2, we can express

direction. The remaining notation is self-explanatory.

Pi,iþ<sup>1</sup> <sup>¼</sup> <sup>K</sup>ð Þ� <sup>i</sup>þ<sup>1</sup> <sup>a</sup> exp <sup>a</sup>iþ<sup>1</sup> � <sup>a</sup>ð Þ� <sup>i</sup>þ<sup>1</sup>

ð Þ� iþ1 <sup>2</sup> <sup>Δ</sup>iþ<sup>1</sup> �

� �<sup>2</sup> � �

Pnð Þ¼ T K<sup>1</sup> exp ð Þ an exp K<sup>2</sup> ¼ K<sup>1</sup> exp ∑

Hence the proof of theorem. q.e.d.

Letting ε go to 0, we have

<sup>1</sup> Δ<sup>i</sup>

¼ ∑ k

<sup>þ</sup> <sup>þ</sup> <sup>a</sup>i<sup>þ</sup> <sup>2</sup> Δ<sup>i</sup> þ � �<sup>2</sup>

¼ P W a

i

In this chapter, we studied some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. We saw how dynamics was decomposed into fast generators k (local Hamiltonians) and slow generators p (couplings) as a Cartan decomposition g ¼ p ⊕ k. Using this decomposition, we used some convexity ideas to completely characterize the reachable set and time optimal control for these

Pnð Þ¼ T K<sup>1</sup> exp T ∑

W a<sup>þ</sup> i � �Δ<sup>þ</sup>

i

<sup>α</sup>kWkð Þ Xd <sup>Δ</sup>iþ<sup>1</sup> � <sup>þ</sup> <sup>o</sup> <sup>Δ</sup>iþ<sup>1</sup> �

<sup>2</sup> are first and second order increments to ai in the positive

<sup>1</sup> <sup>Δ</sup>iþ<sup>1</sup> � � <sup>a</sup>ð Þ� <sup>i</sup>þ<sup>1</sup>

þ W a

ð Þ� iþ1 1 � � � �

� �<sup>2</sup> � � � �

� �<sup>2</sup> � �

<sup>2</sup> <sup>Δ</sup>iþ<sup>1</sup> �

ð Þ� iþ1 <sup>1</sup> <sup>Δ</sup>iþ<sup>1</sup> � <sup>þ</sup> <sup>a</sup>

� �<sup>2</sup> � �

<sup>i</sup> þ W a�

� �

αiWið Þ Xd � �

iþ1 � �Δ� iþ1

K2:

Kð Þ� <sup>i</sup>þ<sup>1</sup> <sup>b</sup> :

ð Þ� iþ1 <sup>2</sup> <sup>Δ</sup>iþ<sup>1</sup> �

2 � � � �

> exp ∑o Δ<sup>2</sup> � � |fflfflfflffl{zfflfflfflffl} # <sup>ε</sup><sup>T</sup>

0 B@

<sup>Δ</sup>iþ<sup>1</sup> � <sup>þ</sup> P W <sup>a</sup>ð Þ� <sup>i</sup>þ<sup>1</sup>

K2:

<sup>Δ</sup>iþ<sup>1</sup> � � �<sup>2</sup>

> 1 CA K2

> > Navin Khaneja Systems and Control Engineering, IIT Bombay, India

\*Address all correspondence to: navinkhaneja@gmail.com

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Nielsen M, Chuang I. Quantum Information and Computation. New York: Cambridge University Press; 2000

[2] Ernst RR, Bodenhausen G, Wokaun A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions. Oxford: Clarendon Press; 1987

[3] Cavanagh J, Fairbrother WJ, Palmer AG, Skelton NJ. Protein NMR spectroscopy. In: Principles and Practice. New York: Academic Press; 1996

[4] Khaneja N, Brockett RW, Glaser SJ. Time optimal control of spin systems. Physical Review A. 2001;63:032308

[5] Khaneja N, Glaser SJ. Cartan decomposition of SU <sup>2</sup><sup>n</sup> ð Þ and control of spin systems. Chemical Physics. 2001; 267:11-23

[6] D'Alessandro D. Constructive controllability of one and two spin 1/2 particles. Proceedings 2001 American Control Conference; Arlington, Virginia; June 2001

[7] Kraus B, Cirac JI. Optimal creation of entanglement using a two qubit gate. Physical Review A. 2001;63:062309

[8] Bennett CH, Cirac JI, Leifer MS, Leung DW, Linden N, Popescu S, et al. Optimal simulation of two-qubit hamiltonians using general local operations. Physical Review A. 2002;66: 012305

[9] Khaneja N, Glaser SJ, Brockett RW. Sub-Riemannian geometry and optimal control of three spin systems. Physical Review A. 2002;65:032301

[10] Vidal G, Hammerer K, Cirac JI. Interaction cost of nonlocal gates. Physical Review Letters. 2002;88:237902 [11] Hammerer K, Vidal G, Cirac JI. Characterization of nonlocal gates. Physical Review A. 2002;66:062321 spin system with triangle topology. Physical Review A;84:062301

[22] Redfield AG. The theory of relaxation processes. Advances in Magnetic and Optical Resonance. 1965;

Physics. 1976;48:199

[23] Lindblad G. On the generators of quantum dynamical semigroups. Communications in Mathematical

[24] Helgason S. Differential Geometry, Lie Groups, and Symmetric Spaces. Cambridge: Academic Press; 1978

[25] Brockett RW. System theory on group manifolds and Coset spaces. SIAM Journal of Control. 1972;10:

[26] Jurdjevic V, Sussmann H. Control systems on lie groups. Journal of Differential Equations. 1972;12:313-329

[27] Jurdjevic V. Geometric Control Theorey. New York: Cambridge

[28] Kostant B. On convexity, the Weyl group and the Iwasawa decomposition. Annales scientifiques de l'cole Normale Supérieure. 1973;6(4):413-455

University Press; 1997

1:1-32

265-284

187

[21] Yuan H, Wei D, Zhang Y, Glaser S, Khaneja N. Efficient synthesis of quantum gates on indirectly coupled spins. Physical Review A;89:042315

DOI: http://dx.doi.org/10.5772/intechopen.80567

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

[12] Yuan H, Khaneja N. Time optimal control of coupled qubits under nonstationary interactions. Physical Review A. 2005;72:040301(R)

[13] Yuan H, Khaneja N. Reachable set of bilinear control systems under time varying drift. System and Control Letters. 2006;55:501

[14] Zeier R, Yuan H, Khaneja N. Time optimal synthesis of unitary transformations in fast and slow qubit system. Physical Review A. 2008;77: 032332

[15] Yuan H, Zeier R, Khaneja N, Lloyd S. Constructing two qubit gates with minimal couplings. Physical Review A. 2009;79:042309

[16] Reiss T, Khaneja N, Glaser S. Broadband geodesic pulses for three spin systems: Time-optimal realization of effective trilinear coupling terms and indirect SWAP gates. Journal of Magnetic Resonance. 2003;165:95

[17] Khaneja N, Glaser S. Efficient transfer of coherence through Ising spin chains. Physical Review A. 2002;66: 060301

[18] Khaneja N, Heitmann B, Spörl A, Yuan H, Schulte-Herbrüggen T, Glaser SJ. Shortest paths for efficient control of indirectly coupled qubits. Physical Review A. 2007;75:012322

[19] Yuan H, Zeier R, Khaneja N. Elliptic functions and efficient control of Ising spin chains with unequal couplings. Physical Review A;77:032340

[20] Yuan H, Khaneja N. Efficient synthesis of quantum gates on a threeConvexity, Majorization and Time Optimal Control of Coupled Spin Dynamics DOI: http://dx.doi.org/10.5772/intechopen.80567

spin system with triangle topology. Physical Review A;84:062301

References

Applied Modern Control

1996

267:11-23

012305

186

Virginia; June 2001

[1] Nielsen M, Chuang I. Quantum Information and Computation. New York: Cambridge University Press; 2000

Principles of Nuclear Magnetic

Oxford: Clarendon Press; 1987

[3] Cavanagh J, Fairbrother WJ, Palmer AG, Skelton NJ. Protein NMR spectroscopy. In: Principles and Practice. New York: Academic Press;

[5] Khaneja N, Glaser SJ. Cartan

[6] D'Alessandro D. Constructive controllability of one and two spin 1/2 particles. Proceedings 2001 American Control Conference; Arlington,

[2] Ernst RR, Bodenhausen G, Wokaun A.

[11] Hammerer K, Vidal G, Cirac JI. Characterization of nonlocal gates. Physical Review A. 2002;66:062321

[12] Yuan H, Khaneja N. Time optimal control of coupled qubits under nonstationary interactions. Physical Review

[13] Yuan H, Khaneja N. Reachable set of bilinear control systems under time varying drift. System and Control

[14] Zeier R, Yuan H, Khaneja N. Time

transformations in fast and slow qubit system. Physical Review A. 2008;77:

[15] Yuan H, Zeier R, Khaneja N, Lloyd S. Constructing two qubit gates with minimal couplings. Physical Review A.

[16] Reiss T, Khaneja N, Glaser S. Broadband geodesic pulses for three spin systems: Time-optimal realization of effective trilinear coupling terms and

indirect SWAP gates. Journal of Magnetic Resonance. 2003;165:95

[17] Khaneja N, Glaser S. Efficient transfer of coherence through Ising spin chains. Physical Review A. 2002;66:

[18] Khaneja N, Heitmann B, Spörl A, Yuan H, Schulte-Herbrüggen T, Glaser SJ. Shortest paths for efficient control of indirectly coupled qubits. Physical Review A. 2007;75:012322

[19] Yuan H, Zeier R, Khaneja N. Elliptic functions and efficient control of Ising spin chains with unequal couplings. Physical Review A;77:032340

[20] Yuan H, Khaneja N. Efficient synthesis of quantum gates on a three-

A. 2005;72:040301(R)

Letters. 2006;55:501

032332

060301

2009;79:042309

optimal synthesis of unitary

Resonance in One and Two Dimensions.

[4] Khaneja N, Brockett RW, Glaser SJ. Time optimal control of spin systems. Physical Review A. 2001;63:032308

decomposition of SU <sup>2</sup><sup>n</sup> ð Þ and control of spin systems. Chemical Physics. 2001;

[7] Kraus B, Cirac JI. Optimal creation of entanglement using a two qubit gate. Physical Review A. 2001;63:062309

operations. Physical Review A. 2002;66:

[9] Khaneja N, Glaser SJ, Brockett RW. Sub-Riemannian geometry and optimal control of three spin systems. Physical

[10] Vidal G, Hammerer K, Cirac JI. Interaction cost of nonlocal gates. Physical Review Letters. 2002;88:237902

Review A. 2002;65:032301

[8] Bennett CH, Cirac JI, Leifer MS, Leung DW, Linden N, Popescu S, et al. Optimal simulation of two-qubit hamiltonians using general local

[21] Yuan H, Wei D, Zhang Y, Glaser S, Khaneja N. Efficient synthesis of quantum gates on indirectly coupled spins. Physical Review A;89:042315

[22] Redfield AG. The theory of relaxation processes. Advances in Magnetic and Optical Resonance. 1965; 1:1-32

[23] Lindblad G. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics. 1976;48:199

[24] Helgason S. Differential Geometry, Lie Groups, and Symmetric Spaces. Cambridge: Academic Press; 1978

[25] Brockett RW. System theory on group manifolds and Coset spaces. SIAM Journal of Control. 1972;10: 265-284

[26] Jurdjevic V, Sussmann H. Control systems on lie groups. Journal of Differential Equations. 1972;12:313-329

[27] Jurdjevic V. Geometric Control Theorey. New York: Cambridge University Press; 1997

[28] Kostant B. On convexity, the Weyl group and the Iwasawa decomposition. Annales scientifiques de l'cole Normale Supérieure. 1973;6(4):413-455

### *Edited by Le Anh Tuan*

This book describes recent studies on modern control systems using various control techniques. The control systems cover large complex systems such as train operation systems to micro systems in nanotechnology. Various control trends and techniques are discussed from practically modern approaches such as Internet of Things, artificial neural networks, machine learning to theoretical approaches such as zero-placement, bang-bang, optimal control, predictive control, and fuzzy approach.

Published in London, UK © 2019 IntechOpen © wacomka / iStock

Applied Modern Control

Applied Modern Control

*Edited by Le Anh Tuan*